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Similarly, Cheeger constant shows how difficult it is to cut the Riemannian manifold into two large pieces [12]. The concept of Cheeger constant in spectral geometry has been incorporated in a very similar way in spectral graph theory. The Cheeger constant of a graph can be bounded above and below by the smallest nontrivial eigenvalue of the Laplacian matrix and normalized Laplacian matrix, respectively, of the graph [14, 29]. Ricci curvature on a graph [6, 23, 27] has been introduced which is analogous to the notion of Ricci curvature in Riemannian geometry [4, 32]. Many results have been proved on manifolds with Ricci curvature bounded below. Lower Ricci curvature bounds have been derived in the context of finite connected graphs. Random walk on graphs is also studied by defining transition probability operator on the same [19]. Eigenvalues of the transition probability operator can be estimated from the spectrum of normalized graph Laplacian [3]. Unlike in a graph, an edge of a hypergraph can be formed with more than two vertices. Thus an edge of a hypergraph is the nonempty subset of the vertex set of that hypergraph [44]. Different aspects of a hypergraph like, Helly property, fractional transversal number, connectivity, chromatic number have been well studied [7, 43]. A hypergraph is used to be represented by an incidence graph which is a bipartite graph with vertex classes, the vertex set and the edge set of the hypergraph and it has been exploited to study Eulerian property, the existence of different cycles, vertex and edge coloring in hypergraphs. A hypergraph can also be represented by a hypermatrix which is a multidimensional array. A recent trend has been developed to explore spectral hypergraph theory using different connectivity hypermatrices. An m-uniform hypergraph on n vertices, where each edge contains the same, m, number of vertices can easily be represented by a hypermatrix of order m and dimension n. In 2005 [33], L. Qi and independently L.H. Lim [26] introduced the concept of eigenvalues of a real supersymmetric tensor (hypermatrix). Then spectral theory for tensors started to develop. Afterward, many researchers analyzed different eigenvalues of several connectivity tensors (or hypermatrices), namely, adjacency tensor, Laplacian tensor, normalized Laplacian tensors, etc. Various properties of eigenvalues of a tensor have been studied in [10, 11, 24, 31, 41, 42, 45, 46]. Using characteristic polynomial, the spectrum of the adjacency matrix of a graph is extended for uniform hypergraphs in [15]. Different properties of eigenvalues of Laplacian and signless Laplacian tensors of a uniform hypergraph have been studied in [20, 21, 22, 34, 35]. Recently Banerjee et al. defined different hypermatrices for non-uniform hypergraphs and studied their spectral properties [2]. Many, but not all properties of spectral graph theory could be extended to spectral hypergraph theory. We also refer to [36] for detailed reading on spectral analysis of hypergraphs using different tensors. One of the disadvantages to study spectral hypergraph theory by tensors (hypermatrices) is the computational complexity to compute the eigenvalues, which is NP-Hard. On the other hand, a simple approach for studying a hypergraph is to represent it by an underlying graph, i.e., by a 2-graph. There are many approaches, such as weighted clique expansion [38, 39, 40], clique averaging [1], star expansion [48]. In this article, we study the spectral properties of the underlying weighted graph of a non-uniform hypergraph by introducing different linear operators (connectivity matrices). This underlying graph cor- responding to a uniform hypergraph is similar as studied by Rodr´ıguez, but the weights of the edges are different. We show that spectrum of these matrices (or operators) can reveal many structural properties of hypergraphs. Connectivity of a hypergraph is also studied by the eigenvalues of these operators. Spectral radii of the same have been bounded by the degrees of a hypergraph. We also bound the diameter of a hypergraph by the eigenvalues of its connectivity matrices. Different properties of a regular hypergraph are characterized by the spectrum. Strong (vertex) chromatic number of a hypergraph is bounded by the eigenvalues. We define Cheeger constant on a hypergraph and show that it can be bounded above and below by the smallest nontrivial eigenvalues of Laplacian matrix and normalized Laplacian matrix,
2 respectively, of a connected hypergraph. We also show an approach to study random walk on a hypergraph that can be performed by analyzing the spectrum of transition probability operator which is defined on that hypergraph. Ricci curvature on hypergraphs is introduced in two different ways. We show that if the Laplace operator, ∆, on a hypergraph satisfies a curvature-dimension type inequality CD(m, K) with m > 1 and K > 0 then any non-zero eigenvalue of ∆ can be bounded below by mK . The spectrum of − m 1 normalized Laplacian operator on a connected hypergraph is also bounded by the Olliviers− Ricci curvature of the hypergraph.
Now we recall some definitions related to hypergraphs. A hypergraph is a pair = (V, E) where V is a set of elements called vertices, and E is a set of non-empty subsetsG of V calledG edges. If all the edges of have the same carnality, m, then is called an m-uniform hypergraph. The rank r( ) and co-rank cr(G ) of a hypergraph are the maximumG and minimum, respectively, of the cardinalitiesG of the edges in .G Two vertices i, j VG are called adjacent if they belong to an edge together, i.e., i, j e for some e E G ∈ ∈ ∈ and it is denoted by i j. Let (V, E) be an∼m-uniform hypergraph with n vertices and let Km be the complete m-uniform G n hypergraph with n vertices. Further let ¯(V, E¯) be the (m-uniform) complement of which is also an G ¯ G m ¯ m-uniform hypergraph such that an edge e E if and only if e / E. Thus the edge set of Kn is E E. A hypergraph (V, E) is called bipartite if V∈ can be partitioned∈ into two disjoint subsets V and V ∪such G 1 2 that for each edge e E, e V1 = and e V2 = . An m-uniform complete bipartite hypergraph is denoted by Km , where∈ V ∩= n 6 and∅ V =∩n . 6 ∅ n1,n2 | 1| 1 | 2| 2 The Cartesian product, , of two hypergraphs =(V , E ) and =(V , E ) is defined by the G1 × G2 G1 1 1 G2 2 2 vertex set V ( 1 2)= V1 V2 and the edge set E( 1 2)= a e : a V1, e E2 e x : e E , x V . Thus,G × G the vertices× (a, x), (b, y) V ( G ×) G are adjacent,({ } × a, x)∈ (b, y∈), if and only×{ if,} either∈ 1 ∈ 2 ∈ G1 × G2 ∼ a = b and x y in , or a b in and x = y. Clearly, if and are two m-uniformS hypergraphs ∼ G2 ∼ G1 G1 G2 with n1 and n2 vertices, respectively, then 1 2 is also an m-uniform hypergraph with n1n2 vertices. For a set S V of a hypergraph (V,G E×), G the edge boundary ∂S = ∂ S is the set of edges in ⊂ G G G with vertices in both S and V S, i.e., ∂S = e E : i, j e, i S and j V S . Similarly the vertex boundary δS for S to\ be the set of all{ vertices∈ in V ∈ S adjacent∈ to some∈ vertex\ } in S, i.e., \ δS = i V S : i, j e E, j S . The Cheeger constant (isoperimetric number) h( ) of a hypergraph (V, E{)∈ is defined\ as ∈ ∈ ∈ } G G ∂S h( ) := inf | | , G φ=S V min(µ(S),µ(V S)) 6 ⊂ \ where µ is a measure on subsets of vertices. Note that, depending on the choice of measure µ we use different tools. For example, if we consider equal weights 1 for all vertices in a subset S then µ(S) becomes the number of vertices in S, i.e., S and combinatorial Laplacian is a better tool to use here. | | On the other hand, if we choose the weight of a vertex equal to its degree, then µ(S) = i S di and normalized Laplacian will be a better choice in this case. Moreover, sometimes for a weighted graph,∈ we P take µ′(∂S)= e (i,j) ∂S wij instead of ∂S in the numerator of h( ), where µ′(∂S) is a measure on the ≡ ∈ | | G set of edges, ∂S, and w is weight of an edge (i, j). P ij In this article we always consider finite non-uniform hypergraph (V, E), i.e., V < , if not mentioned otherwise. G | | ∞
3 2 Adjacency matrix of hypergraphs
The adjacency matrix A = [(A )ij] of a hypergraph =(V, E) is defined as G G G 1 (A )ij = . G e 1 e E, | | − i,jX∈ e ∈ 1 For an m-uniform hypergraph the adjacency matrix becomes (A )ij = dij , where dij is the codegree G m 1 of vertices i and j. The codegree d of vertices i and j is the number of edges− (in ) that contain the ij G vertices i and j both, i.e., d = e E : i, j e . The above definition of adjacency matrix for a uniform ij |{ ∈ ∈ }| hypergraph is similar, but not the same, as defined in [38]. Now, the degree, di, of a vertex i V which n ∈ is the number of edges that contain i can be expressed as di = (A )ij. Now we explore the operator j=1 G form of the adjacency matrix defined above. Let (V, E) be a hypergraph on n vertices. Let us consider a real-valued function f on , i.e., on the vertices ofG , f : V RP. The set of such functions forms a vector G G → n space (or a real inner product space) which is isomorphic to R . For such two functions f1 and f2 on RnG we take their inner product f1, f2 = i V f1(i)f2(i). This inner product space is also isomorphic to . Let us choose a basis = hg ,g ,...,gi ∈ such that g (j) = δ . Now we find the adjacency operator T B { 1 2 n} i ij such that [T ] = A . Here, we also denoteP T by A . Now, our adjacency operator A (which is a linear B G G G operator) is defined as 1 (A f)(i)= f(j). G e 1 j,i j e E, | | − X∼ i,jX∈ e ∈ n It is easy to verify that A f1, f2 = f1, A f2 , for all f1, f2 R , i.e., the operator A is symmetric h G i h G i ∈ G w.r.t. ., . . So the eigenvalues of A are real. Now onwards we shall use the operator and the matrix form G of A hinterchangeably.i G Clearly, for a connected hypergraph the adjacency matrix A , which is real and non-negative (i.e., all G G of its entries are a non-negative real number), possesses a Perron eigenvalue with positive real eigenvector. Moreover, for an undirected hypergraph A is symmetric. The hypergraph , the corresponding weighted G G graph G[A ] (constructed from the adjacency matrix A of ) and the graph G0[A ] have the similar G G G G property regarding graph connectivity and coloring. Here G0[A ] is the underlying unweighted graph of G G[A ]. So, is connected if and only if G[A ] is connected. Thus it is easy to show that a hypergraph G G G is connected if and only if the highest eigenvalue of A is simple and possesses a positive eigenvector G G(see Cor. 1.3.8, [17]). We can also estimate the upper bound for spectral radius ρ(A ) for a connected G hypergraph (V, E) as ρ(A ) max i,j, didj . This equality holds if and only if is a regular hypergraph G G ≤ i j{ } G ∼ (see Cor 2.5,[16]). p Theorem 2.1. Let =(V , E )and =(V , E ) be two m-uniform hypergraphs on n and n vertices, G1 1 1 G2 2 2 1 2 respectively. If λ and µ are eigenvalues of A 1 and A 2 , respectively, then λ + µ is an eigenvalue of A 1 2 . G G G ×G
Proof. Let a, b V1 and x, y V2 be any vertices of 1 and 2, respectively. Let α and β be the eigenvectors corresponding∈ to the∈ eigenvalues λ and µ, respectively.G G Let γ Cn1n2 be a vector with the ∈ entries γ(a, x) = α(a)β(x), where (a, x) [n1] [n2]. Now we show that γ is an eigenvector of A 1 2 ∈ × G ×G
4 corresponding to the eigenvalue λ + µ. Thus,
d(a,x)(b,y) A 1 2 γ(b, y) = γ(b, y) G ×G (a,x),(b,y) m 1 (b,y),(a,x) (b,y) (b,y),(a,x) (b,y) − X∼ X∼ d d = ab α(b)β(x)+ xy α(a)β(y) m 1 m 1 (b,x),(a,x) (b,x) (a,y),(a,x) (a,y) X∼ − X∼ − = β(x)λα(a)+ α(a)µβ(x) = (λ + µ)γ(a, x).
Hence the proof follows. The n-dimensional m-uniform cube hypergraph Q(n, m) consists of the vertex set V = 0, 1, 2,...,m 1 n and the edge set E = x , x ,...,x : x V and x , x differs exactly in one coordinate{ when i =−j } {{ 1 2 m} i ∈ i j 6 } [9]. Note that Q(1, m)= Km and Q(1, m) Q(1, m)= Q(2, m). In general m × n Q(n, m)= Q(1, m) Q(1, m) Q(1, m) . × ×···× z m 2 }| { − 1 1 1 m Since the eigenvalues of AKm are 1, − , − ,..., − , using the above theorem we can easily { zm 1 m }|1 m 1{} find the eigenvalues of AQ(n,m). − − −
2.1 Diameter of a hypergraph and eigenvalues of adjacency matrix A path v v of length l between two vertices v , v V in a hypergraph (V, E) is an alternating 0 − 1 0 1 ∈ G sequence v0e1v1e2v2 ...vl 1elvl of distinct vertices v0, v1, v2,...,vl and distinct edges e1, e2,...,el, such − that, vi 1, vi ei for i = 1,...,l. The distance, d(i, j), between two vertices i, j in a hypergraph is − the minimum∈ length of a i j path. The diameter, diam( ), of a hypergraph (V, E) is the maximumG − G G distance between any pair of vertices in , i.e., diam( ) = max d(i, j): i, j V . Now it is easy to show that the diameter of a hypergraph is lessG than the numberG of{ distinct eigenvalues∈ } of A . G G Theorem 2.2. L et (V, E) be a connected hypergraph with n vertices and minimum edge carnality 3. Let θ be the second largestG eigenvalue (in absolute value) of A . Then G log((1 α2)/α2) diam( ) 1+ − , G ≤ log(λ /θ) max t where λmax is the largest eigenvalue of A with the unit eigenvector X1 = ((X1)1, (X1)2,..., (X1)n) and G α = min (X ) . i{ 1 i}
Proof. A is real symmetric and thus have orthonormal eigenvectors Xl with A Xl = λlXl, where λ1 = G G λmax. Let us choose i, j V such that d(i, j)= diam( ) and r diam( ) be a positive integer. We try ∈ r G ≥ G r to find the minimum value of r such that (A )ij > 0. Using spectral decomposition of A , (A )ij can be G G G
5 express as
n r r t (A )ij = λl (XlXl )ij G l=1 X n λr (X ) (X ) λr(X ) (X ) ≥ max 1 i 1 j − l l i l j l=2 X n 1/2 n 1/2 α2λr θr (X ) 2 (X ) 2 ≥ max − | l i| | l j| Xl=2 Xl=2 α2λr θr(1 α2). ≥ max − − 2 2 r r 2 2 log((1 α )/α ) − Now, (A )ij > 0 if (λmax/θ) > (1 α )/α , which implies that r > log(λ /θ) . Thus the proof follows. G − max Corollary 2.1. Let (V, E) be a k-regular connected hypergraph with n vertices. Let θ be the second largest eigenvalue (in absoluteG value) of A . Then G log(n 1) diam( ) 1+ − . G ≤ log(k/θ)
Remark: For graphs, the above bounds are more sharp [13]. Also note that, in Theorem 2.2, θ = λmax since the underlying graph is not bipartite. 6
2.2 Uniform regular hypergraphs and eigenvalues of adjacency matrices
Let be an m-uniform hypergraph with n vertices and let di be the degree of the vertex i in . Further, let G G n−2 n 1 (m−2) ¯ ¯ ¯ − ¯ m di be the degree of i in . Thus di + di = m 1 . Then A + A = AKn = θ(Jn In), where θ = m 1 , Jn G − G G − − is the (n n) matrix with all the entries are 1 and In is the (n n) identity matrix. If be an m-uniform × ¯ n 1 × G k-regular hypergraph with n vertices then is m− 1 k -regular and A ¯ is a symmetric non-negative G n 1− − G 1 matrix. Thus A ¯ contains a Perron eigenvalue m− 1 k with an eigenvector n. For every non-Perron G − − eigenvector X of A with an eigenvalue λ we have A ¯X = ( θ λ)X. Thus A and A ¯ have the same G G G G eigenvectors. Now we have the following proposition. − −
Proposition 2.1. If is an m-uniform k-regular hypergraph with n vertices, then the minimum eigenvalue G n 1 λn of A satisfies the inequality λn k θ m− 1 . G ≥ − − −
Proof. Let us order the eigenvalues of A as k = λ1 λ2 λn. Then the eigenvalues of A ¯ can be n 1 G ≥ ≥···≥ G ordered as m− 1 k θ λn θ λ2, which implies the result. − − ≥ − − ≥···≥− − 2.3 Hypergraph coloring and adjacency eigenvalues A strong vertex coloring of a hypergraph is a coloring where any two adjacent vertices get different G colors. The strong (vertex) chromatic number γ( ) of a hypergraph is the minimum number of colors needed to have a strong vertex coloring of . G G G Theorem 2.3. Let γ( ) be a hypergraph with r( )= r. Then G G
γ( ) 1+(r 1)λmax(A ). G ≤ − G
6 Proof. For a simple unweighted graph G, the vertex chromatic number χ(G) 1 + λ (G), where ≤ max λmax(G) is the maximum eigenvalue of the adjacency matrix of G [47]. Thus we have γ( )= χ(G0[A ]) G G ≤ 1+ λmax(G0[A ]). Since each element of the adjacency matrix of G0[A ] is less than or equals to the G G same of (r 1)A , we have λmax(G0[A ]) (r 1)λmax(A ). Now the proof follows from the last two G G G inequalities.− ≤ −
Theorem 2.4. Let be a hypergraph with at least one edge. Then G
λmax(A ) γ( ) 1 G . G ≥ − λmin(A ) G Proof. Let k be γ( ). Now A can be partitioned as G G
0 A 12 ... A 1 G G k A 21 0 ... A 2k A = G. . . G. . G . . .. . A k1 A k2 ... 0 G G Using the Lemma 3.22 in [5] we have λmax(A )+(k 1)λmin(A ) 0. Since has at least one edge, G − G ≤ G λmin(A ) < 0. Hence the proof follows. G From Theorems 2.3 and 2.4 we have the following corollary.
Corollary 2.2. Let be a hypergraph with at least one edge. Then G
λmin(A ) 1/(r( ) 1). | G | ≥ G − 3 Combinatorial Laplacian matrix and operator of a hyper- graph
Now we define our (combinatorial) Laplacian operator L for a hypergraph (V, E) on n vertices. We G G 2 take the same usual inner product f1, f2 = i V f1(i)f2(i) for the n dimensional Hilbert space L ( ) h i ∈ G constructed with all real-valued functions f on , i.e., f : V R. Now our Laplacian operator PG → L : L2( ) L2( ) G G → G defined as 1 1 (L f)(i) := (f(i) f(j)) = dif(i) f(j). G e 1 − − e 1 j,i j e E, | | − j,i j e E, | | − X∼ i,jX∈ e X∼ i,jX∈ e ∈ ∈ It is easy to verify that L is symmetric (self-adjoint) w.r.t. the usual inner product ., . , i.e., L f1, f2 = GRn h i h 1G i f1, L f2 , for all f1, f2 . So the eigenvalues of L are real. Since L f, f = i j e E, e 1 (f(i) h G i ∈ G h G i ∼ i,j∈ e | |− − ∈ f(j))2 0, for all f Rn, L is nonnegative, i.e., the eigenvalues of L are nonnegative.P P The Rayleigh G G Quotient≥ (f) of a∈ function f : V R is defined as RLG → 1 2 i j e E, e 1 (f(i) f(j)) L f, f ∼ i,j∈ e | |− − G ∈ LG (f)= h i = 2 . R f, f P P i V f(i) h i ∈ P 7 For standard basis we get the matrix from Laplacian operator L as G
di if i = j, 1 − if i j, (L )ij = e E, e 1 G i,j∈ e | |− ∼ ∈ 0P elsewhere. So, L = D A , where D is the diagonal matrix where the entries are the degrees di of the vertices i G G G G of . Any λ(L− ) R becomes an eigenvalue of L if, for a nonzero u Rn, it satisfies the equation G G ∈ G ∈ L u = λ(L )u. (1) G G
Let us order the eigenvalues of L as λ1(L ) λ2(L ) λn(L ). Now find an orthonormal 2 G G ≤ G ≤ ··· ≤ G basis of L ( ) consisting of eigenfunctions of L , uk,k = 1,...,n as follows. First we find u1 from the G G expression λ1(L ) = infu L2( ) L u,u : u = 1 . Now iteratively define Hilbert space of all real- G G valued functions on with∈ theG {h scalar producti || || ., . ,}H := v L2( ): v,u = 0 for l k . Then G h i k { ∈ G h li ≤ } we start with the function u1 (eigenfunction for the eigenvalue λ1(L )) and find all the eigenvalues of G LG u,u LG u,u L as λk(L ) = infu H 0 h i . Thus, λ2(L ) = inf0=u u1 h i . We can also find λn(L ) as G G ∈ k−{ } u,u G 6 ⊥ u,u G h i h i λn(L ) = supu L2( ) L u,u : u =1 . G ∈ G {h G i || || } Gerˇsgorin circle theorem [18] provides a trivial bounds on eigenvalues λ of L as λ 2dmax. As in G | | ≤ m adjacency matrix for an m-uniform hypergraph with n vertices we have L +L ¯ = LKn = φm(n)In θJn, n−2 G G G − n n 1 (m−2) where φ (n) = − , θ = , J is the (n n) matrix with all the entries 1 and I is the m n 1 m 1 m 1 n × n (n n) identity matrix.− − Now it is easy− to show that for an m-uniform hypergraph with n vertices, if × G 0= λ1 λ2 λn be the eigenvalues of L with the corresponding eigenvectors, 1 n = X1,X2,...,Xn, ≤ ≤···≤ G respectively, then the eigenvalues of L ¯ are 0 = λ1,φm(n) λ2,...,φm(n) λn with the same set of G − − corresponding eigenvectors X1,X2,...,Xn, respectively. Now we have the following computations on eigenvalues. The eigenvalues of L m are 0 and φ (n) with the (algebraic) multiplicity 1 and n 1, Kn m − respectively. The eigenvalues of L m are 0,φ (n + n ),φ (n + n ) φ (n ) and φ (n + n ) φ (n ) Kn1,n2 m 1 2 m 1 2 − m 1 m 1 2 − m 2 with the multiplicity 1, 1, n 1 and n 1, respectively. If 0,λ ,...,λ and 0,µ ,...,µ are the 1 − 2 − 2 n1 2 n2 eigenvalues of L 1 and L 2 , respectively, where 1 and 2 are two m-uniform hypergraphs with number of G G G G vertices, n1 and n2, respectively, then the eigenvalues of L 1+ 2 are 0,φm(n1 +n2), φm(n1+n2) φm(n1)+λ2, G G ..., φ (n + n ) φ (n )+ λ , φ (n + n ) φ (n )+ µ , ..., φ (n + n ) φ (n )+− µ , where m 1 2 − m 1 n1 m 1 2 − m 2 2 m 1 2 − m 2 n2 + = ¯ ¯ . Also note that the Theorem 2.1 holds for Laplacian matrices of two uniform hypergraphs G1 G2 G1 ∪ G2 and using it we can compute the eigenvalues of LQ(n,m). The definition of Laplacian matrix for a uniform hypergraph is similar, but not the same as defined in [38, 39, 40] for studying spectral properties of uniform hypergraphs. In [38], Rodr´ıguez studied the Laplacian eigenvalues of a uniform hypergraph and several metric parameters such as the diameter, mean distance, excess, cutsets and bandwidth. A very trivial upper bound for diameter by distinct eigenvalues of Laplacian matrix is mentioned. The bounds on parameters related to distance in (uniform) hypergraphs, such as eccentricity, excess were investigated in [39] . In [40], the distance between two vertices in a (uniform) hypergraph was explained by the degree of a real polynomial of Laplacian matrix. The relation between parameters related to partition-problems of a (uniform) hypergraph with second smallest and the largest eigenvalues of Laplacian matrix was explored. Only the lower bound for isoperimetric number and bipartition width were figured out in terms of the second smallest eigenvalue of the same. Upper bound for max cut, domination number and independence number was mentioned by the largest Laplacian-eigenvalue.
8 3.1 Hypergraph connectivity and eigenvalues of a (combinatorial) Laplacian matrix
n Now it is easy to verify that is connected iff λ2(L ) = 0 and then any constant function u R G G 6 ∈ becomes the eigenfunction with the eigenvalue λ1(L )=0. If (V, E) has k connected components, then G G the (algebraic) multiplicity of the eigenvalue 0 of L is exactly k. So, we call λ2(L ) algebraic weak G G connectivity of hypergraph . The following theorems show more relation of λ2(L ) with the different G G aspects of connectivity of hypergraph . A set of vertices in a hypergraph G is a weak vertex cut of if weak deletion of the vertices from that set increases the number of connectedG components in . TheGweak connectivity number κ ( ) is the G W G minimum size of a weak vertex cut in . G Theorem 3.1. Let (V, E) be a connected hypergraph with n( 3) vertices, such that, contains at least G ≥ G one pair of nonadjacent vertices and dmax cr( ). Then λ2(L ) κW ( ). ≤ G G ≤ G Proof. Let W be a weak vertex cut of such that W = κ ( ). Let us partition the vertex set of as G | | W G G V1 W V2 such that no vertex in V1 is adjacent to a vertex in V2. Let V1 = n1 and V2 = n2. ∪Since∪ is connected, u is constant. Let us construct a real-valued function| | u, orthogonal| | to u , as G 1 1 n if i V , 2 ∈ 1 u(i)= 0 if i W , n if i ∈ V . − 1 ∈ 2 Since any vertex j W is adjacent to a vertex in V1 and also to a vertex in V2, and dmax cr( ), then ∈ ≤ G 1 dij cr( ) 1 for all j W and i / W . Now, for any vertex i / W , we define ki = j W,j i e E, e 1 ≤ G − ∈ ∈ ∈ ∈ ∼ i,j∈ e | |− ≤ ∈ dij 1 j W,j i cr( ) 1 κW ( ). Thus, for all i V1, (L u)(i) = diu(i) j V1,j i Pe E, e 1 uP(j) = n2di ∈ ∼ G − ≤ G ∈ G − ∈ ∼ i,j∈ e | |− − ∈ 2 nP2(di ki) = n2ki. Similarly, for all i V2, we have (L u)(i) = n1ki.P Hence, λP2(L ) u L u,u − ∈ G − G || || ≤ h G i ≤ n n2κ ( )+ n2n κ ( )= κ ( ) u 2. 1 2 W G 1 2 W G W G || || Theorem 3.2. Let (V, E) be a hypergraph on n vertices. Then, for a nonempty S V , we have G ⊂ λn(L ) S V S cr( ) 1 λ2(L ) S V S (r( ) 1) G | || \ | ∂S G − G | || \ |. G − n ≥| | ≥ r( )2/4 n ⌊ G ⌋ Proof. Let us construct a real-valued function u, orthogonal to u1, as n n if i S, u(i)= − S ∈ n if i V S, − S ∈ \ where n = S . Now we have S | | 1 1 1 ∂S n2 L u,u = (u(i) u(j))2 ∂S r( )2/4 n2. | |r( ) 1 ≤h G i e 1 − ≤| |cr( ) 1⌊ G ⌋ G − i j e E, | | − G − X∼ i,jX∈ e ∈ The inequality on the right side holds because if i S and j V S, then the number of terms in the parentheses in the above equation is maximum when∈e = r( )∈ and there\ are equal number of vertices in e from S and V S, respectively, and is equal to r( )2/4| |. Similary,G the inequality on the left side holds when \ ⌊ G ⌋ 2 there is only one vertex from a partition in e ∂S. Thus, λ2(L ) u = λ2(L )nnS(n nS) L u,u 1 2 2 ∈ 2 G || || G − ≤h G 1 i ≤2 ∂S r( ) /4 n . Similarly, we have λn(L ) u = λn(L )nnS(n nS) L u,u ∂S n . | | cr( ) 1 ⌊ G ⌋ G || || G − ≥ h G i≥| | r( ) 1 HenceG the− proof follows. G −
9 Now we bound the Cheeger constant
∂S h( ) := min | | G φ=S V min( S , V S ) 6 ⊂ | | | \ | of a hypergraph (V, E) from below and above through λ2(L ). G G Theorem 3.3. Let (V, E) be a connected hypergraph with n vertices. Then G cr( ) 1 h( ) 2λ2(L ) G − . G ≥ G r( ) r( ) 1 G G − Proof. Let S be a nonempty subset of V , such that, h( ) = ∂S / S and S V . Let us define a real-valued function u as G | | | | | |≤| | 1 if i S, √ S u(i)= | | ∈ ( 0 elsewhere.
P tS (e) L u,u e∈∂S h G i Let us define tS(e) = v : v e S, e E and t(S) = ∂S . Now we have u 2 = |{ ∈ ∩ ∈ }| | | || || 1 2 r( ) 1 2 r( ) 1 tS (e) G − G − e E i j, e 1 (u(i) u(j)) e ∂S i e S cr( ) 1 u(i) = cr( ) 1 e ∂S S . Thus we have λ2(L ) ∈ i,j∼ e | |− − ≤ ∈ ∈ ∩ G − G − ∈ | | G ≤ r( ) 1 ∂S{ t(S}∈) r( ) 1 ∂S¯ t(S¯) PG − P| | P G −P| | ¯ P cr( ) 1 S and similarly λ2(L ) cr( ) 1 S¯ , where S = V S. Now from these two inequalities we G − | | G ≤ G − | | \ r( ) r( ) 1 G G − have λ2(L ) h( ). Thus the proof follows. G ≤ 2(cr( ) 1) G G − Theorem 3.4. Let (V, E) be a connected hypergraph on n vertices. If d is the maximum degree of G max G and λ2 = λ2(L ) then G h( ) < (r( ) 1) (2d λ )λ . G G − max − 2 2
Proof. Let u2 be the eigenfunction with the eigenvaluepλ2, such that, u2 = 1. Let φ = S V , such that ∂S || || 6 ⊂ | | R h( )= S and S V /2. Let u : V be a function defined by G | | | |≤| | → u(i)= u (i) for all i V . | 2 | ∈ Thus, 1 1 λ > (u(i) u(j))2 d (u(i) u(j))2 =: M (say). (2) 2 e 1 − ≥ r( ) 1 ij − i j e E, | | − G − i j X∼ i,jX∈ e X∼ ∈ The rest of the proof is similar to the proof for graphs [29]. Equation (2), by using Cauchay-Schwarz inequality, implies that
2 2 2 1 i j dij u (i) u (j) M ∼ | − | . (3) 2 ≥ r( ) 1 P i j dij(u(i)+ u(j)) G − ∼ Now proceed in a similar way as in the proof givenP in [29]. Let t < t < < t be all different 0 1 ··· h values of u(i), i V . For k = 0, 1,...,h, let us define V := i V : u(i) t , and we denote δ (e) = ∈ k { ∈ ≥ k} k
10 min Vk e , (V Vk) e for each edge e ∂Vk and δ(Vk) = mine ∂V δk(e) . Let δ( ) = mink [h] δ(Vk) , ∈ k ∈ where{| [h∩]=| |1, 2\,...,h∩ |}. Now ∈ { } G { } { } d u2(i) u2(j) = d u2(i) u2(j) ij| − | ij − i j i j X∼ u(iX)∼u(j) ≥ h 2 2 2 2 2 2 = dij tk tk 1 + tk 1 tl+1 + tl+1 tl − − − −···− − k=1 i j ∼ X u(Xi)=tk u(j)=tl
uV2 ,pt(L )uV1 = pt(0)a1b1 + pt(λi(L ))aibi h G i G i=2 Xn V1 V2 | || | pt(λi(L ))aibi ≥ n − G p i=2 X n n V V | 1|| 2| (1 λ)t a2 b2 (6) ≥ n − − v i i p u i=2 i=2 uX X V V t(n V )(n V ) = | 1|| 2| (1 λ)t −| 1| −| 2| . (7) p n − − p n n 2 The inequality (6) follows from Cauchy-Schwarz inequality, whereas the equality (7) holds since i=2 ai = 2 2 n V1 n 2 n V2 u ( u , f ) = −| | , and b = −| | . V1 V1 1 n i=2 i n P || The|| − inequality|h i| in (6) is strict. This is because the equality in Cauchy-Schwarz inequality holds if and only if a = cb , for all i, for some constantP c. However, it is possible only when V = V or V = V V , i i 1 2 1 \ 2 √ V1 V2 t √(n V1 )(n V2 ) which is not the case here. So, we get uV2 ,pt(L )uV1 > | || | (1 λ) −| | −| | . Now, if we h G i n − − n choose (n V1 )(n V2 ) −| | −| | log V1 V2 t | || | , ≥ log qλn(LG )+λ2(LG ) λn(L ) λ2(L ) G − G 12 uV2 ,pt(L )uV1 becomes strictly positive. Thus the proof follows. h G i Corollary 3.1. For a connected hypergraph (V, E) on n vertices and with at least one pair of nonadjacent G vertices, log(n 1) diam( ) − . G ≤ &log λn(LG )+λ2(LG ) ' λn(L ) λ2(L ) G − G Corollary 3.2. Let (V, E) be an connected hypergraph on n vertices and with at least one pair of non- G adjacent vertices. Then, for any S V , we have ⊂ δS 1 (1 λ)2 | | (n S ) − − , S ≥ −| | (1 λ)2(n S )+ S | | − −| | | | where λ = 2λ2(LG ) and δS = j V S : d(i, j)=1, for some i S is the vertex boundary of S. λn(LG )+λ2(LG ) { ∈ \ ∈ } Moreover, if S n/2 then we have | | ≤
δS 2λn(L )λ2(L ) G G | | 2 2 . S ≥ λn(L ) + λ2(L ) | | G G Proof. Take V = S, V = V S δS and t = 1. Then, using Equation (7) of Theorem 3.6, we have 1 2 \ −
V1 V2 (n V1 )(n V2 ) 0= uV2 ,pt=1(L )uV1 > | || | (1 λ) −| | −| | . h G i p n − − p n 2 Since V V2 = S δS, this implies (1 λ) (n S )( S + δS ) > S (n S δS ). Thus the first part of the result\ follows.∪ − −| | | | | | | | −| |−| | Now, when S n S , from the above inequality we get | | ≤ −| | 2 2 δS 1 (1 λ) 1 (1 λ) 2λn(L )λ2(L ) G G | | 2− − − −2 = 2 2 . S ≥ (1 λ) + S /(n S ) ≥ (1 λ) +1 λn(L ) + λ2(L ) | | − | | −| | − G G
3.3 Bounds on λ2(L ) and λn(L ) of a hypergraph G G G Let e i , i ,...,i E be any edge in . Then, for e and u Rn, we define a homogeneous polynomial ≡{ 1 2 m} ∈ G ∈ of degree 2 in n variables by
e e e | | | | | | 2 2 1 L (e)u := u(ij) u(ij)u(ik). G − e 1 j=1 | | − j=1, k=1, X Xij e ijX=i e ∈ 6 k∈
2 Thus, L u,u = e E L (e)u . h G i ∈ G Theorem 3.7. LetP (V, E) be a connected hypergraph on n(> 2) vertices. Then G
di1 + di2 + + di|e| e λ2(L ) min ··· −| | : e i1, i2,...,i e E . G ≤ e ≡{ | |} ∈ | |
13 n Proof. Let e i1, i2,...,i e E be any edge in . Now, let us construct a function u R , as ≡{ | |} ∈ G ∈ 1/2 e − if i e, u(i)= |0| elsewhere.∈ 2 1 1 e 1 | | Now, λ2(L ) e E L (e)u = e (di1 + di2 + + di|e| ) e 1 2 2 e . Thus the proof follows. G ≤ ∈ G | | ··· − | |− · · | | Corollary 3.3.PLet (V, E) be a connected hypergraph with n(> 2) vertices. Then G
di1 + di2 + + di|e| e λn(L ) min ··· −| | : e i1, i2,...,i e E . G ≥ e ≡{ | |} ∈ | | Theorem 3.8. Let (V, E) be an m(> 2)-uniform connected hypergraph with n vertices. Then G 2 2 2di(m 1) 1+ 4(m 1) dimiDmax 2di(m 1)+1 λn(L ) max − − − − − : i V , G ≤ 2(m 1) ∈ ( p − ) where mi =( j,j i dj)/di(m 1) is the average 2-degree of the vertex i and Dmax = max dxy : x, y V . ∼ − { ∈ } Proof. Let unPbe an eigenfunction of L with the eigenvalue λn(L ) (= λn( ), say). Then we have G G G n d (u (i) u (j))2 =(m 1)λ ( ) u (i)2. (8) ij n − n − n G n i,j,i j i=1 X∼ X
From the eigenvalue equation of L (i.e., from (1)) for the vertex i we have (m 1)(di λn(L ))un(i) G − − G ≤ j,j i Dmaxun(j). Using Lagrange identity and summing both sides over i we get ∼ P n n n 2 2 2 2 2 2 2 (m 1) (di λn(L )) un(i) (m 1) diDmax un(j) Dmax(un(j) un(k)) (9) − − G ≤ − − − i=1 i=1 j,j i i=1 1 j n (m 1)2(d λ ( ))2 (m 1)2d m D2 +(m 1)λ ( ) u (i)2 0. (10) − i − n G − − i i max − n G n ≤ i=1 X Hence there exists a vertex i for which we have (m 1)(d λ ( ))2 (m 1)d m D2 + λ ( ) 0. − i − n G − − i i max n G ≤ This provides our desired result. 14 Corollary 3.4. Let (V, E) be an m(> 2)-uniform connected hypergraph on n vertices. Then G 2 2 2 2dmax(m 1) 1+ 4(m 1) dmax E 2dmin(m 1)+1 λn(L ) − − − | | − − , G ≤ 2(m 1) p − where d and d are the maximum and the minimum degrees, respectively, of . max min G 2 Proof. Since dimi = ( j,j i dj)/(m 1) dmax and Dmax E the result follows from the above ∼ − ≤ ≤ | | theorem. P 1 Another upper bound of λn(L ) can also be found from the Theorem 5 in [37] as λn(L ) maxi j di+ G G ≤ 2 ∼ 1 n dj + m 1 k i,k≁j dik + k j,k≁i djk + k i,k j dik djk . − ∼ ∼ ∼ ∼ | − | P P P o 4 Normalized Laplacian matrix and operator of a hypergraph Now we define normalized Laplacian operator and matrix ∆ for a hypergraph (V, E) on n vertices. G G Let µ be a natural measure on V given by µ(i) = di. We consider the inner product f1, f2 µ := 2 2 h i R i V µ(i)f1(i)f2(i), for the n dimensional Hilbert space l (V,µ), given by l (V,µ) = f : V . Now∈ our normalized Laplacian operator ∆ : l2(V,µ) l2(V,µ) is defined as { → } P G → 1 1 1 1 (∆ f)(i) := (f(i) f(j)) = f(i) f(j). (11) G di e 1 − − di e 1 j,i j e E, | | − j,i j e E, | | − X∼ i,jX∈ e X∼ i,jX∈ e ∈ ∈ It is easy to verify that the eigenvalues of ∆ are real and nonnegative, since f1, ∆ f2 µ = ∆ f1, f2 µ, Rn 1 G 2 Rhn G i h G i for all f1, f2 and ∆ f, f µ = m 1 i j dij(f(i) f(j)) 0, for all f . The Rayleigh Quotient ∈ h G i − ∼ − ≥ ∈ ∆ (f) of a function f : V R is defined as R G → P 1 2 i j e E, e 1 (f(i) f(j)) ∆ f, f µ ∼ i,j∈ e | |− − G ∈ ∆G (f)= h i = 2 . (12) R f, f µ P P i V dif(i) h i ∈ For standard basis we get the matrix form of normalized LaplacianP operator ∆ as G 1 if i = j, 1 1 − e E, if i j, (∆ )ij = di e 1 (13) G i,j∈ e | |− ∼ ∈ 0P elsewhere. 1 So, ∆ = I R , where R = A D− is normalized adjacency matrix, which is a row-stochastic matrix. G G G G R can be considered− as a probabilityG transition matrix of a random walk on . G G Now we order the eigenvalues of ∆ as λ1(∆ ) λ2(∆ ) λn(∆ ) and find an orthonormal 2 G G ≤ G ≤ ···≤ G basis of l (V,µ) consisting of eigenfunctions of ∆ , uk,k = 1,...,n, as we did it for Laplacian operator. G The expression λ1(∆ ) = infu l2(V,µ) 0 ∆ u,u µ : u = 1 provides u1 and λ1(∆ ). The rest of G ∈ −{ }{h G i || || } G ∆G u,u µ the eigenvalues are iteratively estimated from the expression λk(∆ ) = infu 0 h i , where G ∈Hk−{ } u,u µ h i 2 ∆G u,u µ := v l (V,µ): v,u = 0 for l k .λ (∆ ) can also be expressed as λ (∆ ) = inf 1 h i . k l µ 2 2 u u u,u µ H { ∈ h i ≤ } G G ⊥ h i We can also define normalized Laplacian operator (and matrix) on a hypergraph (V, E) on n vertices G as follows. Here, we consider the usual inner product f1, f2 := i V f1(i)f2(i), for the n dimensional h i ∈ P 15 Hilbert space L2(V ) constructed with all real-valued functions f : V R and the other normalized Laplacian operator : L2(V ) L2(V ) and is defined as → LG → 1 1 1 1 ( f)(i) := (f(i) f(j)) = f(i) f(j). (14) LG d d e 1 − − d d e 1 i j j,i j e E, | | − i j j,i j e E, | | − X∼ i,jX∈ e X∼ i,jX∈ e p ∈ p ∈ For standard basis we get the matrix form of the above normalized Laplacian operator as LG 1 if i = j, 1 1 − e E, if i j, ( )ij = d d e 1 (15) G √ i j i,j∈ e | |− ∼ L ∈ 0P elsewhere. Two normalized Laplacian operators in (11) and (14) are equivalent. Hence, the matrices in (13) and (15) are similar and thus have the same spectrum. In this article we use the normalized Laplacian operator defined in (11) and its matrix form1 in (15). It is easy to verify that the eigenvalues of ∆ for an m(> 2)- G unform hypergraph lie in [0, 2) and the number of connected components in is equal to the (algebraic) multiplicity of eigenvalue 0. When is connected, u is constant. Many theoremsG for normalized Laplacian G 1 matrix can be constructed similar to the theorems for Laplacian matrix (operator). We see that λ2(∆ ) G can also bound the Cheeger constant defined as, ∂S h( ) := inf | | G φ=S V min(vol(S),vol(S¯)) 6 ⊂ for a hypergraph (V, E) from below and above. Here vol(S)= i S di. This Cheeger constant h( ) can G ∈ G also be bounded above and bellow by λ2(∆ ), respectively, as follows. G P Theorem 4.1. Let (V, E) be a connected hypergraph on n vertices. Then G cr( ) 1 2λ2(L ) G − h( ) < (r( ) 1) (2 λ2)λ2. G r( ) r( ) 1 ≤ G G − − G G − p Proof. The proof is similar to the proofs of Theorem 3.3 and Theorem 3.4. A similar thoerem as Theorem 3.5 can be written for λ2(∆ ) as G Theorem 4.2. For a connected hypergraph on n vertices, G 4 diam( ) . G ≥ n(r( ) 1)dmaxλ2(∆ ) G − G Theorem 3.6 also holds for the respective eigenvalues of ∆ as G 1 Note that, two non-isomorphic hypergraphs of order m > 2 may have the same normalized Laplacian matrix ∆G (or the normalized adjacency matrix RG ). For an example, It happens when all the 2-element subsets of the vertex set of the hypergraph are subsets of a fixed number of edges. For instance, existence of a (combinatorial) simple incomplete 2-design on the vertex set of a hypergraph where each edge is considered as a block. A particular example is the Fano plane, which is a finite projective plane of order 2 with 7 points, represents a 3-uniform hypergraph on 7 vertices with the vertex set 1, 2, 3, 4, 5, 6, 7 and the edge set 1, 2, 3 , 1, 4, 7 , 1, 5, 6 , 2, 4, 6 , 2, 5, 7 , 3, 4, 5 , 3, 6, 7 , where each pair of vertices{ belongs to exactly} one edge. Fano{{ plane} is{ a regular} { balance} d{ incomplete} { block} { (7, 3, }1)-design.{ }} Thus, the normalized 3 Laplacian (adjacency) matrices for Fano plane and K7 , respectively, are the same. 16 Theorem 4.3. Let (V, E) be a connected hypergraph with n vertices and at least one pair of nonadjacent vertices. Then, for VG ,V V such that V = V = V V , we have 1 2 ⊂ 2 6 1 6 \ 2 vol(V¯1)vol(V¯2) log vol(V1)vol(V2) d(V1,V2) . ≤ &log qλn(∆G )+λ2(∆G ) ' λn(∆ ) λ2(∆ ) G − G Then we have the similar corollary as we have for L . G Corollary 4.1. For a connected hypergraph (V, E) with n vertices and at least one pair of nonadjacent G vertices, (n 1)dmax log( − ) diam( ) dmin . G ≤ &log λn(∆G )+λ2(∆G ) ' λn(∆ ) λ2(∆ ) G − G Moreover, if is regular then G log(n 1) diam( ) − . G ≤ &log λn(∆G )+λ2(∆G ) ' λn(∆ ) λ2(∆ ) G − G Now we find bounds on eigenvalues of δ . It is easy to verify that for a hypergraph containing at G G least one pair of nonadjacent vertices λ2(∆ ) 1 λn(∆ ). A similar theorem of Theorem 3.8 can be G G stated as follows. ≤ ≤ Theorem 4.4. Let (V, E) be an m-uniform connected hypergraph with n vertices. Then G 2 2 2(m 1)di 1+ 1 4(m 1)di + 4(m 1) diDmaxmi λn(∆ ) max − − − − − : i V , G ≤ 2(m 1)d ∈ ( p − i ) where mi =( j,j i dj)/di(m 1) is the average 2-degree of the vertex i and Dmax = max dxy : x, y V . ∼ − { ∈ } Proof. Let unPbe an eigenfunction of ∆ with the eigenvalue λn(∆ ). Then, from the eigenvalue equation G G for λn(∆ ) and un, we have G n 2 2 dij(un(i) un(j)) =(m 1)λn(∆ ) diun(i) . (16) − − G i,j,i j i=1 X∼ X n Rest of the proof is similar to the proof of Theorem 3.8. Now the expression in (10) becomes i=1 (m 2 2 2 − 1)(1 λn(∆ ) )di (m 1)miDmax + λn(∆ ) un(i) 0. Using similar arguments as in Theorem 3.8 we − G − 2 − G ≤ 2 P have (m 1)diλn(∆ ) +(1 2(m 1)di)λn(∆ )+(m 1)(di miDmax) 0, which proves the theorem. − G − − G − − ≤ When m = 2, becomes a triangulation and the above upper bound coincides with the result proved in [25] for a triangulation.G Now we have the following corollary. Corollary 4.2. For an m-uniform connected hypergraph (V, E) with the maximum and the minimum G degrees dmax and dmin, respectively, we have 2 2 2 2(m 1)dmax 1+ 1 4(m 1)dmin + 4(m 1) dmax E λn(∆ ) − − − − − | | . G ≤ 2(m 1)d p − min 17 5 Random walk on hypergraphs A random walk on a hypergraph (V, E) can be considered as a sequence of vertices v0, v1,...,vt and it can be determined by the transitionG probabilities P (u, v) = Prob(x = v x = u) which is independent i+1 | i of i. Thus, a simple random walk on a hypergraph (V, E) is a Markov chain, where a Markov kernel on V is a function G P ( , ): V V [0, + ), · · × → ∞ such that y V P (x, y)=1 x V . Here P (x, y) is called reversible if there exists a positive function µ( ) on the state space∈ V , such that∀ ∈P (x, y)µ(x)= P (y, x)µ(y). A random walk is reversible if its underlying· Markov kernelP is reversible. It is easy to see that a random walk on a connected hypergraph with co-rank greater than 2 is ergodic, i.e., P is (i) irreducible: i.e., for all x, y V , P t(x, y) > 0 for some t N and (ii) aperiodic: i.e., g.c.d t : P t(x, y) = 1. We can consider the∈ transition probabilities P (x,∈ y) for a G connected hypergraph (V,{ E) with cr}( ) > 2 as G G 1 P e E, |e|−1 x,y∈ e P (x, y)= ∈d if x y, G x ∼ 0 else. Now, let us consider P : l2(V,µ) l2(V,µ) as a transition probability operator for the random walk on G → . Thus ∆ = I P , where I is the identical operator in l2(V,µ). Hence, for an eigenvalue λ(∆ ) of ∆ , G G − G G G we always get an eigenvalue (1 λ(∆ )) of P . Let αk =1 λk(∆ ) be the eigenvalues of P of an m(> 2) − G G − G G hypergraph on n vertices, for k = 0,...,n. Then, 1 < αn αn 1 α2 α1 = 1. Hence P 1, − G since the SpecG P ( 1, 1]. Let us consider the powers− P t of P≤ for t ≤ Z+ ≤as composition of|| operators.|| ≤ G G Below we recall the⊂ theorem− for convergence of random walkG on graphs∈ [19] in the context of connected hypergraphs (V, E) with cr( ) > 2 on n vertices. G G Theorem 5.1. For any function f l2(V,µ), take ∈ 1 f = d f(i). vol(V ) i i V X∈ Then, for any positive integer t, we have P t f f ρt f , || G − || ≤ || || where ρ = maxk=1 1 λk(∆ ) = max( 1 λ2(∆ ) , 1 λn(∆ ) ) is the spectral radius of P and f = 6 | − G | | − G | | − G | G || || f, f µ. Consequently,h i p P t f f 0 || G − || → as t , i.e., P t f converges to a constant f as t . →∞ G →∞ 1 t Thus, after t 1 ρ log(1/ǫ) steps P f f becomes less than ǫ f . We define the equilibrium ≥ − || G − || || || transition probability operator P : l2(V,µ) l2(V,µ) as G → 1 P u(i)= dju(j). G vol(V ) j V X∈ Thus, P f = f, for all functions f l2(V,µ). Using the above theorem we find that P t converges to P G ∈ G G as t . We→∞ also refer our readers to [28] where a set of Laplacians for hypergraphs have been defined to study high-order random walks on hypergraphs. 18 6 Ricci curvature on hypergraphs Here we discuss two aspects of Ricci curvature on hypergraphs. Let us recall our transition probability operator 1 e E, e 1 x,y∈ e | |− P (x, y)= ∈ G P dx for a hypergraph (V, E). Clearly P (x, y) is reversible. Let us define the Laplace operator G G ∆ := ∆ − G which is also acting on l2(V,µ). Thus ∆ = P I and for any f l2(V,µ) we have (∆f)(i) = 1 1 G − ∈ e E, (f(j) f(i)). Now we discuss two aspects of Ricci curvature in the sense of Bakry and di j,i j e 1 ∼ i,j∈ e | |− − ∈ EmeryP [4]P and Ollivier [32]. For graphs, readers may also see [6, 23, 27]. 6.1 Ricci curvature on hypergraphs in the sense of Bakry and Emery Let us define a bilinear operator Γ: l2(V,µ) l2(V,µ) l2(V,µ) × → as 1 Γ(f , f )(i) := ∆(f (i)f (i)) f (i)∆f (i) f (i)∆f (i) . 1 2 2{ 1 2 − 1 2 − 2 1 } Then the Ricci curvature operator, Γ2, is defined as 1 Γ (f , f )(i) := ∆Γ(f , f )(i) Γ(f , ∆f )(i) Γ(f , ∆f )(i) . 2 1 2 2{ 1 2 − 1 2 − 2 1 } Now, for our hypergraph we have G 1 1 1 Γ(f, f)(i)= (f(i) f(j))2. 2 d e 1 − i j,j i e E, | | − X∼ i,jX∈ e ∈ Then, from the proof of Theorem 1.2 in [27], we can express our Ricci curvature operator on a hypergraph as G 1 1 1 1 1 2 Γ2(f, f)(i)= (f(i) 2f(j)+ f(k)) 4 di dj e 1 e 1 − j,j i e E, | | − k,k j e E, | | − X∼ i,jX∈ e X∼ j,kX∈ e ∈ ∈ 1 1 1 1 1 1 2 (f(i) f(j))2 + (f(i) f(j)) . − 2 di e 1 − 2 di e 1 − j,j i e E, | | − j,j i e E, | | − X∼ i,jX∈ e X∼ i,jX∈ e ∈ ∈ We have omitted the variable i in the above equation. For simplicity, we do the same for the following equations which hold for all i V . Let m and K be the dimension∈ and the lower bound of the Ricci curvature, respectively, of Laplacian operator ∆. Then we say that ∆ satisfies curvature-dimension type inequality CD(m, K) for some m > 1 if 1 Γ (f, f) (∆f)2 + KΓ(f, f) for all f l2(V,µ). 2 ≥ m ∈ 19 If Γ KΓ, then ∆ satisfies CD( , K). Any connected (finite) hypergraph (V, E) satisfies CD(2, 1 1), 2 d∗ ≥ ∞ 1 G − where d = supi V supj i(di)/( e E, e 1 ). ∗ ∈ ∼ i,j∈ e | |− ∈ Now, Theorem 2.1 in [6] canP be stated in the context of hypergraphs as follows. Theorem 6.1. If ∆ satisfies a curvature-dimension type inequality CD(m, K) with m > 1 and K > 0 then mK λ2(∆ ) . G ≥ m 1 − 6.2 Ricci curvature on hypergraphs in the sense of Ollivier The Ollivier’s Ricci curvature (also known as Ricci-Wasserstein curvature) is introduced on a separable and complete metric space (X,d), where each point x X has a probability measure px( ). Let us denote the structure by (X,d,p). Let (µ, ν) be the set of probability∈ measures on X X projecting· to µ and ν. Now ξ (µ, ν) satisfies C × ∈ C ξ(A X)= µ(A),ξ(X B)= ν(B), A, B X. × × ∀ ⊂ Then the transportation distance (or Wasserstein distance) between two probability measures µ, ν on a metric space (X,d) is defined as 1(µ, ν) := inf d(x, y)dξ(x, y). T ξ (µ,ν) X X ∈C Z × Now on (X,d,p), the Ricci curvature of (X,d,p) for distinct x, y X is defined as ∈ (p ,p ) κ(x, y) := 1 T1 x y . − d(x, y) For a connected hypergraph (V, E) we take d(x, y) = 1 for two distinct adjacent vertices x, y and we consider the probability measureG 1 1 e E, , if y x, dx e 1 x,y∈ e | |− ∼ px(y)= ∈ ( 0, P otherwise, for all x V . Now Theorem 3.1 in [6] also holds for a connected hypergraph as follows ∈ Theorem 6.2. Let be a connected hypergraph on n vertices. Then κ λ2(∆ ) λn(∆ ) 2 κ, G G where the Ollivier’s RicciG curvature of is at least κ. ≤ ≤ ≤ − G As in [23], one can also introduce a scalar curvature (suggested in Problem Q in [32]) for a vertex x in as G 1 κ(x) := κ(x, y). d x y,x y X∼ Acknowledgements The author is sincerely thankful to Richard Buraldi for inspiring him to introduce connectivity matrix on a hypergraph. The author is very grateful to Saugata Bandyopadhyay and Asok Nanda for scrutinizing the manuscript. The author is also thankful to J¨urgen Jost, Shipping Liu, Satyaki Mazumder, Shibananda Biswas, Sushil Gorai, Shirshendu Chowdhury, Swarnendu Datta and Amitesh Sarkar for fruitful discus- sions. 20 References [1] S. Agarwal ; Jongwoo Lim ; L. Zelnik-Manor ; P. Perona ; D. Kriegman ; S. Belongie, Beyond pairwise clustering, In 2005 IEEE Computer Society Conference on Computer Vision and Pattern Recognition (CVPR’05), 2:838-845, 2005. [2] A. Banerjee, A. Char, B. Mondal. Spectra of general hypergraphs. 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