Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. 8 (1997), 64{71. DOUBLY STOCHASTIC GRAPH MATRICES Russel l Merris Let L(G) b e the n n Laplacian matrix of graph G: This note intro duces 1 the p ositive de nite doubly sto chastic matrix (G)= I +L(G) : n Let G =(V; E) b e a (simple) graph with vertex set V = fv ;v ;:::;v g and 1 2 n edge set E of cardinality o(E )=m: Let D (G) b e the diagonal matrix whose (i; i)- entry is d(v ); the degree of vertex v : Denote by A(G) the n n adjacency matrix i i whose (i; j )-entry is 1 if fv ;v g2E; and 0 otherwise. The Laplacian matrix is i j L(G)=D(G)A(G):By Gersgorin's Theorem, L(G) is p ositive semide nite. Because its rows sum to zero, it is a singular M -matrix. Denote the eigenvalues of L(G)by =0: Then, either from Kirchhoff's Matrix- 1 2 n1 n Tree Theorem [2; 16] or the general theory of M -matrices [1, p.156; 10], > 0 n1 if and only if G is connected, if and only if L(G) is irreducible. This observation led Fiedler [7] to de ne a(G)= ; calling it the algebraic connectivity of G: n1 Numerous variations on the theme if \inverting" L(G)have app eared in the recent literature. (See, for example, [9], [13], [14], [17], and [21].) The purp ose of this note is to intro duce another. 1. De nition. If G isagraph on n vertices, de ne 1 I +L(G) (G)= : n 2. Prop osition. If G isagraph on n vertices, then (G) is a positive de nite doubly stochastic matrix that is entrywise positive if and only if G is connected. Pro of. Because M (G)=I +L(G) is a symmetric p ositive de nite matrix, it is n invertible and its inverse is p ositive de nite. Indeed, the eigenvalues of (G) are 1 1 > 0: (1) 1 1+ a(G) 1+ 1 Because M (G)isanM-matrix, its inverse is entrywise nonnegative. Because M (G) is irreducible if and only if G is connected [8, Chap. 3], it follows from the general 0 1991 Mathematics Sub ject Classi cation: Primary, 05C50. Secondary, 15A48, 15A51, 15A57. 64 Doubly sto chastic graph matrices 65 theory of M -matrices [1, p.141] that (G)isentrywise p ositive if and only if G is connected. Let e b e the n-by-1 matrix each of whose entries is 1. Because e is an eigen- vector of L(G) corresp onding to =0;it is an eigenvector of (G) corresp onding n to the Perron eigenvalue 1. (Indeed, the eigenvectors of (G) and L(G) are iden- tical.) Hence, (G)e = e; proving that (G)isrow sto chastic. Beacuse (G)is symmetric, the pro of is complete 2 Recall that a symmetric matrix is doubly p ositive (nonnegative) if it is en- trywise p ositive (nonnegative) and p ositive (semi)de nite. Thus, (G) is a doubly nonnegative, doubly sto chastic matrix which is doubly p ositive if and only if G is connected. It turns out that doubly nonnegative, doubly sto chastic matrices have b een the sub ject of several recent articles (see, e.g., [11] and [19]); M -matrices whose inverses are sto chastic were studied in [20]. Moreover, (G) is not the rst doubly sto chastic matrix to b e asso ciated with G: If is the largest vertex degree 1 of (a nontrivial graph) G; then I L(G) is doubly sto chastic. n Denote by G _ u the graph on n +1 vertices and m + n edges obtained from G by joining each of its vertices to a new vertex u: (A vertex adjacenttoevery other vertex of a graph is said to b e dominating vertex.) Observe that I + L(G) n is the n n principal submatrix of L(G _ u) obtained by deleting the row and column corresp onding to u: Let us take a moment to consider Prop osition 2 from this p ersp ective. Let H b e a xed but arbitrary graph on n +1 vertices, fv ;v ;:::;v g: 1 2 n+1 Denote by L (H ) the principal submatrix of L(H ) obtained by deleting row and i column i: (It seems that n+1 P L (H ) i i=1 is known in the chemical literature as an \Eichinger matrix" [15].) A real matrix P is said to b e doubly superstochastic if there exists a doubly sto chastic matrix S such that every elementofP is greater than or equal to the cor- resp onding elementofS: Evidently, for a nonnegative matrix to b e sup ersto chastic each of its row and column sums must b e at least 1. As the matrix 0 1 0 0 1 @ A 0 0 1 A = 1 1 0 shows, however, this necessary condition is not sucient. Indeed [4], a nonnegative n n matrix is doubly sup ersto chastic if and only if the sum of the elements of every p q submatrix is at least p + q n: 3. Prop osition. Let H bea connectedgraph on n +1 vertices, fv ;v ;:::;v g: 1 2 n+1 1 Then L (H ) isapositive de nite doubly superstochastic matrix. It is doubly i stochastic if and only if v is a dominating vertex. It is entrywise positive if and i only if the graph H v (obtained by deleting vertex v and al l the edges incident i i with it) is connected. 66 Russell Merris Pro of. Because H is connected, L (H ) is p ositive de nite. Hence, it is invertible i ^ and its inverse is p ositive de nite. Let H b e the graph obtained from H by adding ^ edges, if necessary, so that v b ecomes a dominating vertex. then L (H )=D+ i i L(H); where D is an n n matrix whose only nonzero entries (if any) are 1's i ^ on the diagonal. In particular, eachentry of the M -matrix B = L (H ) is bigger i than or equal to the corresp onding entry of the M -matrix A = L (H ): It follows i 1 1 from the general theory of M -matrices [10; 12, Thm 2.5.4] that A B (in the entrywise sense), with equality if and only if D =0:Hence, the result is a consequence of Prop osition 2 and the fact that A is irreducible if and only if H v i is connected. 2 1 If H is a tree and d(v )= 1;then eachentry of A = L (H ) is a p ositive i i integer. In particular, A S for every n n doubly sto chastic matrix S: 1 A general graph{theoretic interpretation for the entries of L (H ) can b e i obtained from the \all minors Matrix-Tree Theorem" [2]. Sp eci cally, the (r;s)- entry of the classical adjoint adj L (H ) is the numb er of 2-tree spanning forests of i H in which v and v lie in one tree and v in the other. (If r = s and " = fv ;v g r s i r i is an edge of H; this is the numb er of spanning trees of H that contain "; an observation rst made in [16, Thm. 2].) Of course, det L (H ) = t(H ); the i numb er of spanning trees in H: 4. Corollary. Suppose H is a connectedgraph on n +1 vertices. Fix i and let 0 be the eigenvalues of L (H ): Then 1 with equality if 1 2 n i n and only if v is a dominating vertex. i Pro of. 1= is the largest eigenvalue of the doubly sup ersto chastic matrix n 1 L (H ) : 2 i Wenow return to (G): 5. De nition. Let G =(V; E) beagraph with vertex set V = fv ;v ;:::;v g 1 2 n and edge set E: Denote by F the family consisting of al l disconnectedspanning G forests of G: For each F 2F ; let (F ) be the product of the numbers of vertices G in the connectedcomponents of F and (F ) the product of the numbers of vertices i in the connectedcomponents of F that do not contain vertex v : Final ly, de ne i F (i; j )=fF 2F : v and v belong to the same component of F g: G G i j 6. Prop osition. Suppose G is a graph with n 2 vertices and t(G) spanning trees. Then the (i; j )-entry of (G) is a fraction whose numerator is P (2) t(G)+ (F ); i F2F (i;j ) G and whose denominator is P (3) nt(G)+ (F ): F2F G Pro of. Let H = G _ u; vhere u is viewed as the (n +1)st vertex of H: Then, as wehave observed, L (H )=I +L(G): Thus, (G) = adj L (H ) =t(H ): n+1 n n+1 Doubly sto chastic graph matrices 67 To each spanning tree of G there corresp ond n di erent spanning trees of H (obtained by joining u to each of the n vertices of G). Thus, of the spanning trees T of H; exactly nt(G)have the prop erty that T u is connected. If T is a spanning tree of H such that F = T u is a disconnected spanning forest of G; then u can/must b e adjacentinT to exactly one vertex in each comp onentofF: This gives (F ) p ossibilities for T: Hence, t(H ) is given byFormula (3).