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A signed graph Σ = (G, σ) is an underlying graph G =(V, E) with a signature function σ : E 1, 1 . Two types of signed distances matrices are introduced → { − } by Hameed et al. [3]. In this paper, we define two signed distance laplacian matrices for signed graphs and characterize balance in signed graphs using these matrices. The signed distance laplacian spectra of some classes of unbalanced signed graphs are also studied.
First we recall the definition of signed distances and corresponding signed distance matrices defined in [3]. Given a signed graph Σ = (G, σ), the sign of a path P in Σ is defined as σ(P )= ∈ σ(e) . The shortest path between two given vertices u Qe E(P ) and v is denoted by P and the collection of all shortest paths P by ; (u,v) (u,v) P(u,v) and d(u, v) denotes the usual distance between u and v .
Definition 1.1 (Signed distance matrices [3]). Auxiliary signs are defined as:
(S1) σ (u, v)= 1 if all shortest uv -paths are negative, and +1 otherwise. max − (S2) σ (u, v) = +1 if all shortest uv -paths are positive, and 1 otherwise. min − Signed distances are:
(d1) d (u, v)= σ (u, v)d(u, v) = max σ(P u,v ): P u,v u,v d(u, v). max max { ( ) ( ) ∈ P( )} (d2) d (u, v)= σ (u, v)d(u, v) = min σ(P ): P d(u, v). min min { (u,v) (u,v) ∈ P(u,v)} And the signed distance matrices are:
max (D1) D (Σ) = (dmax(u, v))n×n .
min (D2) D (Σ) = (dmin(u, v))n×n .
Definition 1.2 ([3]). Two vertices u and v in a signed graph Σ are said to be distance-compatible (briefly, compatible) if dmin(u, v) = dmax(u, v) . And Σ is said to be (distance-)compatible if every two vertices are compatible. Then Dmax(Σ) = Dmin(Σ) = D±(Σ).
The two complete signed graphs from the distance matrices Dmax and Dmin is defined as follows. Roshni, Germina, Hameed, Zaslavsky 3
max Definition 1.3 ([3]). The associated signed complete graph KD (Σ) with respect to Dmax(Σ) is obtained by joining the non-adjacent vertices of Σ with edges having signs
σ(uv)= σmax(uv).
min The associated signed complete graph KD (Σ) with respect to Dmin(Σ) is obtained by joining the non-adjacent vertices of Σ with edges having signs
σ(uv)= σmin(uv).
The distance laplacian matrix of a graph was introduced and studied by Mustapha et al. in [1]. We introduce two signed distance laplacian matrices for signed graphs as follows.
The transmission Tr(v) of a vertex v is defined to be the sum of the distances from v to all other vertices in G . That is, Tr(v)= ∈ d(v,u) . The transmis- Pu V (G) sion matrix Tr(G) for a graph is the diagonal matrix with diagonal entries Tr(vi).
Definition 1.4. Now we define two signed distance laplacian matrices for signed graphs as
(L1) Lmax(Σ) = Tr(G) Dmax(Σ). − (L2) Lmin(Σ) = Tr(G) Dmin(Σ). − When Σ is compatible, Lmax(Σ) = Lmin(Σ) = L±(Σ) .
2 Balanced signed graphs
To study balance in signed graphs using signed distance laplacian matrices, it is necessary to study the laplacian matrix of weighted signed graphs. Let Σ=(G, σ) be a signed graph and let w be a positive weight function defined on the edges of Σ . We denote a weighted signed graph by (Σ,w) . For a weighted signed graph (Σ,w), w(Σ) is the product of all the weights given to the edges of Σ . We use the 4 Signed Distance Laplacian Matrices for Signed Graphs notation u v when the vertices u and v are adjacent and similar notation for ∼ the incidence of an edge on a vertex.
Definition 2.1. Let (Σ,w) be a weighted signed graph. Its adjacency matrix
A(Σ,w)=(aij)n is defined as the square matrix of order n = V (G) where | | σ(vivj)w(vivj) if vi vj aij = ∼ 0 otherwise. Definition 2.2. For a weighted signed graph (Σ,w) , its weighted laplacian matrix is defined as L(Σ,w) = D(Σ,w) A(Σ,w) where the diagonal matrix D(Σ,w) is − diag ∼ w(e) . The matrix D(Σ,w) is the weighted degree matrix of (Σ,w). Pe:vi e
To discuss oriented incidence matrix, we orient the edges (in an arbitrarily but
fixed way) of the weighted signed graph. For an oriented edge ~ej = −−→vivk we take vi as the tail of that edge and vk as its head and we write t(~ej)= vi and h(~ej)= vk . We choose weights w from the set of positive real numbers and take √w as the positive square root of w .
Definition 2.3. Given a weighted signed graph (Σ,w) , its (oriented) weighted incidence matrix is defined as H(Σ,w)=(ηviej ) where
σ(ej) w(ej) if t(~ej)= vi, p ηviej = w(ej) if h(~ej)= vi, −p 0 otherwise. T ′ Let H (Σ,w) = (ηeivj ) be the transpose of the weighted incidence matrix ′ H(Σ,w) . Thus, ηeivj = ηvj ei .
Theorem 2.4. For a weighted signed graph (Σ,w) , L(Σ,w) = H(Σ,w)HT (Σ,w).
Proof. Let v1, v2,...,vn and ~e1, ~e2,...,~em be the vertices and edges in Σ , respec- th T m ′ tively. The (i, j) entry of HH is ηviek ηe v . Pk=1 k j Roshni, Germina, Hameed, Zaslavsky 5
′ 2 For i = j , ηv e η = η = 0 if and only if ek is incident to vi . i k ekvj viek 6 ′ 2 T Then ηv e η = ( w(ek)) = w(ek) . Thus, the diagonal entry in HH is i k ekvj ±p ∼ w(e). Pe:vi e ′ For i = j , ηv e η = 0 if and only if ek is an edge joining vi and vj . Then 6 i k ekvj 6 ′ ηv e η = σ(ek)w(ek). i k ekvj − In both cases, the (i, j)th entry of HHT coincides with the (i, j)th entry of L(Σ,w) , hence the proof.
Lemma 2.5. For a weighted signed tree (Σ,w) , det L(Σ,w)=0.
Proof. A tree on n vertices has n 1 edges. Thus, H(Σ,w) is a matrix of order − n (n 1) and hence L(Σ,w) = H(Σ,w)HT (Σ,w) has rank less than n . This × − implies det L(Σ,w)=0.
Lemma 2.6. Let (Σ,w) be a weighted signed graph where the underlying graph is a cycle Cn of order n . Then det L(Σ,w)=2w(Cn)(1 σ(Cn)) . −
Proof. Let the cycle be Cn = v ~e v ~e v ~e vn− ~en− vn~env . The weighted inci- 1 1 2 2 3 3 ··· 1 1 1 dence matrix H(Σ,w) is
σ(e ) w(e ) 0 ... 0 w(en) 1 p 1 −p w(e ) σ(e ) w(e ) ... 0 0 1 2 2 −p . p. . . . ...... . 0 0 ... σ(en−1) w(en−1) 0 p 0 0 ... w(en− ) σ(en) w(en) −p 1 p Expanding along the first row to find the determinant we get
n det H(Σ,w)= σ(e ) w(e )M , +( 1) w(en)M ,n 1 p 1 1 1 − p 1 6 Signed Distance Laplacian Matrices for Signed Graphs where
σ(e ) w(e ) ... 0 0 2 2 p...... M1,1 = det , 0 ... σ(en− ) w(en− ) 0 1 p 1 0 ... w(en−1) σ(en) w(en) −p p
w(e ) σ(e ) w(e ) ... 0 1 2 2 −p . p...... M1,n = det . 0 0 ... σ(en− ) w(en− ) 1 p 1 0 0 ... w(en−1) −p
Since M1,1 and M1,n are determinants of triangular matrices,
det H(Σ,w)= σ(e ) w(e )σ(e ) w(e ) σ(en) w(en) 1 p 1 2 p 2 ··· p n +( 1) w(en)( w(e )) ( w(en− )) − p −p 1 ··· −p 1 =(σ(Cn) 1) w(Cn). − p Now, det HT (Σ,w) = det H(Σ,w), hence,
det L(Σ,w)=(σ(Cn) 1) w(Cn).(σ(Cn) 1) w(Cn) − p − p =2w(Cn)(1 σ(Cn)). − Lemma 2.7. Let (Σ,w) be a signed graph, where the underlying graph is a unicyclic graph of order n with unique cycle C . Then det L(Σ,w)=2w(Σ)(1 σ(C)). −
Proof. Define the orientation of edges so that for i < j the edge ~ei,j has tail vi and head vj . Let C = v ~e v ~e vp~epv be the unique cycle, and label the vertices so 1 1 2 2 ··· 1 that each edge ~eij not in C has vi nearer to C than vj ; in other words, the vertex labels increase when moving away from C . Then the incidence matrix H(Σ,w) has Roshni, Germina, Hameed, Zaslavsky 7 the following form:
H(C,w) ∗ w e ... ( p+1) −p ∗ ∗ ∗ 0 w(e ) ... p+2 −p ∗ ∗ , O 0 0 ... ∗ ∗ . . . . . ...... 0 0 ... 0 w(en) −p which is an upper-triangular block matrix whose first diagonal block is H(C,w) and whose other diagonal elements correspond to the heads of edges not in C . Hence,
n det H(Σ,w) = det H(C,w) ( w(e)) · Y −p k=p+1 =(σ(C) 1)( 1)n−p w(Σ). − − p Thus,
det L(Σ,w) = det H(Σ,w) det HT (Σ,w) =2w(Σ)(1 σ(C)). − Lemma 2.8. Let (Σ,w) be a signed graph whose underlying graph is a 1 -forest.
Then det L(Σ,w)= w(Σ) 2(1 σ(Cψ)) , where the product runs over all compo- Qψ − nent 1 -trees ψ having unique cycle Cψ.
Proof. By suitable reordering of vertices and edges we can make the laplacian L(Σ,w) into a block diagonal matrix, where the blocks correspond to the com- ponents of the 1 -forest;. Thus, det L(Σ,w) is the product of the laplacian deter- minants of the components. The components are 1 -trees. By using Lemma 2.7 we get the expression for the determinant.
A signed graph is said to be contrabalanced if it contains no positive cycles [5]. 8 Signed Distance Laplacian Matrices for Signed Graphs
Lemma 2.9. Let L(Σ,w) be a weighted signed graph having n vertices and Ψ be a spanning subgraph of (Σ,w) having exactly n edges. Then, det L(Ψ,w) = 0 if 6 and only if Ψ is a contrabalanced 1 -forest.
Proof. A spanning subgraph Ψ of n edges that is not a 1 -forest must have a com- ponent that has fewer edges than vertices, that is, it is a tree T , and det L(T,w)=0 by Lemma 2.5. Since det L(Ψ,w) is the product of the laplacians of the components of (Ψ,w) , det L(Ψ,w) = 0 if Ψ is not a 1 -forest.
Now, assuming (Ψ,w) is a 1-forest, by Lemma 2.8, det L(Ψ,w) = 0 if and 6 only if Ψ contains no positive cycles, which means that Ψ is a contrabalanced 1 -forest.
Let c(Ψ) denote the number of components of a signed graph Ψ .
Theorem 2.10. Let (Σ,w) be a signed graph; then
det L(Σ,w)= 4c(Ψ)w(Ψ), X Ψ where the summation runs over all contrabalanced spanning 1 -forests Ψ of G .
Proof. Since L(Σ,w) = H(Σ,w)HT (Σ,w) , by the Binet–Cauchy theorem [2] we get
det L(Σ,w)= det H(J) det HT(J)= det L(J, w), X X J J where J is a spanning subgraph of G with exactly n edges.
Thus by Lemma 2.9, det L(Σ,w) = det L(Ψ) where the summation runs PΨ over all contrabalanced spanning 1 -forests Ψ of G. Since every cycle Cψ in Ψ is negative, the factor 1 σ(Cψ) = 2 . That gives the formula of the theorem. −
Recall that we assume the signed graph Σ is connected.
Theorem 2.11. A weighted signed graph (Σ,w) is balanced if and only if the de- terminant of its weighted laplacian matrix is equal to 0. Roshni, Germina, Hameed, Zaslavsky 9
Proof. In the formula of Theorem 2.10 every term is positive, so the determinant is 0 if and only if there is no contrabalanced 1 -forest in Σ . Since Σ is connected, that implies it has no negative circle, that is, it is balanced.
We recall the characterization theorem for balance in signed graphs using signed distances proved by Hameed et al. in [3].
Theorem 2.12 ([3]). For a signed graph Σ the following statements are equivalent:
(i) Σ is balanced.
max (ii) The associated signed complete graph KD (Σ) is balanced.
min (iii) The associated signed complete graph KD (Σ) is balanced.
± (iv) Dmax(Σ) = Dmin(Σ) and the associated signed complete graph KD (Σ) is balanced.
Now we are ready to characterize balance in signed graphs using signed distance laplacian matrices.
Theorem 2.13. The following properties of a signed graph Σ are equivalent.
(i) Σ is balanced.
(ii) The max-signed distance laplacian determinant det Lmax(Σ) = 0.
(iii) The min-signed distance laplacian determinant det Lmin(Σ) = 0.
(iv) Lmax(Σ) = Lmin(Σ) and det L±(Σ) = 0.
max Proof. Corresponding to the associated signed complete graph KD (Σ) , we define Dmax a weighted signed complete graph (K (Σ),w) where w(e) = dmax(u, v) and Dmin (K (Σ),w) , where w(e)= dmin(u, v) for an edge e = uv . Then 10 Signed Distance Laplacian Matrices for Signed Graphs
max min L(KD (Σ),w)= Lmax(Σ) and L(KD (Σ),w)= Lmin(Σ) .
Thus, by Theorem 2.11,
max det Lmax(Σ) = 0 if and only if KD (Σ) is balanced, and
min det Lmin(Σ) = 0 if and only if KD (Σ) is balanced.
Hence, by Theorem 2.12 we get the required characterization.
3 Signed distance laplacian spectrum
Now, we move to the signed distance laplacian spectrum of some types of signed graph.
Theorem 3.1. A signed graph Σ is balanced if and only if Lmax(Σ) = Lmin(Σ) = L±(Σ) and L±(Σ) is cospectral with L(G) .
Proof. Suppose Σ = (G, σ) is balanced. Then Σ can be switched to an all positive signed graph Σζ =(G, +) . Now, D±(Σ) exists [3] and is similar to D±(Σζ ), that is, D±(Σζ )= SD±(Σ)S−1 [3]. So,
SL±(Σ)S−1 = S(Tr(G) D±(Σ))S−1 = Tr(G) D±(Σζ ) − − = L±(Σζ ).
Thus, L±(Σ) is similiar to L(G) , which implies that L±(Σ) is cospectral with L(G).
Conversely, suppose Lmax(Σ) = Lmin(Σ) = L±(Σ) and L± is cospectral with L(G) . Thus, det L±(Σ) = det L(G) = 0 and hence, by Theorem 2.13, Σ is bal- anced. Roshni, Germina, Hameed, Zaslavsky 11
A signed graph Σ is t -transmission regular if Tr(v) = ∈ d(v,u) = t for Pu V (G) all v V (G). Odd cycles C k are k(k + 1) -transmission regular and even cycles ∈ 2 +1 2 C2k are k -transmission regular.
Theorem 3.2. If the signed graph Σ is t -transmission regular, then the signed distance laplacian eigenvalues of Lmax(Σ)(or Lmin(Σ)) are t λ , where λ is an − eigenvalue of Dmax(Σ)(or Dmin(Σ)) .
− For the odd unbalanced cycle Cn , where n =2k + 1 , there is a unique shortest max − min − ± − path between any two vertices. Thus, L (Cn ) = L (Cn ) = L (Cn ) . The signed distance spectrum of an odd unbalanced cycle is given in [3]. Thus, we get, − the signed distance laplacian spectrum of Cn as an immidiate corollary.
− Theorem 3.3. For an odd unbalanced cycle Cn , where n =2k +1 , the spectrum of L± is
k j 2 kπ k 1 ( 1) k( 1) sin ((2j + 1) 2n ) k(k + 1) k( 1) − − k(k + 1) − π 2 π − − − 2 − sin((2j + 1) 2n ) − sin ((2j + 1) 2n ) . 1 2(j =0, 1, 2,...,k 1) −
Acknowledgement
The first author would like to acknowledge her gratitude to Department of Science and Technology, Govt. of India for the financial support under INSPIRE Fellowship scheme Reg No: IF180462. The second author would like to acknowledge her grat- itude to Science and Engineering Research Board (SERB), Govt. of India, for the financial support under the scheme Mathematical Research Impact Centric Support (MATRICS), vide order no.: File No. MTR/2017/000689. 12 Signed Distance Laplacian Matrices for Signed Graphs
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