On the inverse matrix of the Laplacian and all ones matrix Sho Suda (Joint work with Michio Seto and Tetsuji Taniguchi) International Christian University JSPS Research Fellow PD November 21, 2012 Sho Suda (International Christian Univ.) On the inverse matrix of L + J Nov. 21, 2012 1 / 16
Contents ▶ Why do we consider ( L + J ) − 1 ? ▶ Main results on K = ( L + J ) − 1 : bounds on entries of K and characterization of graphs which attain our bounds. ▶ Related work on doubly stochastic graph matrices Ω = ( L + I ) − 1 . ▶ Further problems. Sho Suda (International Christian Univ.) On the inverse matrix of L + J Nov. 21, 2012 2 / 16
Laplacian matrix ▶ G = ( V, E ) : an undirected finite graph with no loop and no multiple edge, that is V is a finite set and E is a subset in {{ x, y } : x, y ∈ V, x ̸ = y } . ▶ G is connected if, for any two distinct vertices in G , there exists a path from one to the other. ▶ The Laplacian matrix L of G is defined to be d i if i = j, L i,j = − 1 if { i, j } ∈ E, 0 otherwise . ▶ J : the all ones matrix. Sho Suda (International Christian Univ.) On the inverse matrix of L + J Nov. 21, 2012 3 / 16
Laplacian matrix ▶ G : a graph with n vertices. ▶ L : the Laplacian matrix of a graph G . ▶ λ 1 ≤ · · · ≤ λ n : all the eigenvalues of L . ▶ It is well known that λ 1 = 0 with an eigenvector 1 and λ 2 > 0 if and only if G is connected. In this talk, we consider K := ( L + J ) − 1 for a connected graph. Sho Suda (International Christian Univ.) On the inverse matrix of L + J Nov. 21, 2012 4 / 16
Reproducing kernel Hilbert spaces . Definition . A vector space H is called a reproducing kernel Hilbert space over some set X if 1. H is a Hilbert space consisting of functions on X , 2. for any x in X , there exists a non-zero function k x in H such that f ( x ) = ⟨ f, k x ⟩ H for any function f in H , where ⟨· , ·⟩ H denotes the inner product of H . . ▶ { k x : x ∈ X } : the set of reproducing kernel. ▶ We call K = ( ⟨ k x , k y ⟩ ) x,y ∈ X the Gram matrix of H . Sho Suda (International Christian Univ.) On the inverse matrix of L + J Nov. 21, 2012 5 / 16
Reproducing kernel Hilbert spaces over graphs ▶ G = ( V, E ) : a connected graph with Laplacian matrix L . ▶ Take X = V and F as the set of all real valued functions on V . Consider the following inner product ∑ ∑ v ( x )) + uLv T ⟨ u, v ⟩ = ( u ( x ))( x ∈ V x ∈ V for u and v in F . ▶ Then the Gram matrix of this Hilbert space is given by K = ( L + J ) − 1 . Sho Suda (International Christian Univ.) On the inverse matrix of L + J Nov. 21, 2012 6 / 16
An example ▶ Let G = ( V, E ) be a path of length 5 with V = { 1 , · · · , 5 } and E = {{ i, i + 1 } : 1 ≤ i ≤ 4 } . ▶ Then L and K are given as follows: 1 − 1 0 0 0 − 1 2 − 1 0 0 L = 0 − 1 2 − 1 0 , 0 0 − 1 2 − 1 0 0 0 − 1 1 31 11 − 4 − 14 − 19 11 16 1 − 9 − 14 K = 1 − 4 1 11 1 − 4 . 25 − 14 − 9 1 16 11 − 19 − 14 − 4 11 31 Sho Suda (International Christian Univ.) On the inverse matrix of L + J Nov. 21, 2012 7 / 16
Properties of Gram matrix K = ( L + J ) − 1 ▶ It is easy to see that each row and column sum is 1 n . ▶ Define r i,j := K i,i + K j,j − 2 K i,j . ▶ d ( i, j ) denotes the path-length distance between i and j . . Proposition (Klein and Randi´ c 1993) . Let G be a connected graph with n vertices, Laplacian matrix L and Gram matrix K = ( L + J ) − 1 . Then the following hold: 1. { r i,j : 1 ≤ i, j ≤ n } satisfies the axiom of distance, 2. r i,j ≤ d ( i, j ) with equality if and only if there is the unique path between i and j . . Sho Suda (International Christian Univ.) On the inverse matrix of L + J Nov. 21, 2012 8 / 16
Properties of Gram matrix K = ( L + J ) − 1 ▶ Define K = K ( G ) to be the maximum value on the diagonal entries of K for G . . Theorem 1 (Seto-S-Taniguchi) . Let G be a connected graph with n vertices, K the Gram matrix of G . Then 1 n ≤ K ≤ K ( P n ) with left equality if and only if G is the complete graph and with right equality if and only if G is the path. . Sketch of the proof: 1. Let G ′ be a graph obtained by deleting an edge of G with Gram matrix K ′ . Then, for any j , K l,l ≥ K ′ l,l . 2. Show that for a tree G if K i,i is minimum then i must be a leaf. 3. Let G ′ be a subtree of G obtained by deleting a leaf 1 with Gram matrix K ′ . Then K 1 , 1 = ( n − 1) 2 2 , 2 + ( n − 1) 2 K ′ where 2 is the unique n 2 n 2 vertex adjacent to 1 in G . Sho Suda (International Christian Univ.) On the inverse matrix of L + J Nov. 21, 2012 9 / 16
Properties of Gram matrix K = ( L + J ) − 1 ▶ Define K = K ( G ) to be the minimum value on the entries of K for G . . Theorem 2 (Seto-S-Taniguchi) . Let T be a tree with n vertices, K the Gram matrix of T . Then K ( P n ) ≤ K ≤ 0 with left equality if and only if T is the path and with right equality if and only if T is the star. . Sketch of the proof: 1. Generally it holds that K ( G ) ≤ 0 with equality iff the corresponding vertex is a dominating vertex. 2. Show that if K i,j is minimum then i and j must be leaves. 3. Let T ′ be a subtree of G obtained by deleting leaves 1 , n with Gram matrix K ′ . Then K 1 ,n = ( n − 2) 2 n 2 d T ′ (2 , n − 1) − 2( n − 1) K ′ 2 ,n − 1 − n − 1 n 2 n 2 where 2 , n − 1 are the unique vertices adjacent to 1 , n in T respectively. Sho Suda (International Christian Univ.) On the inverse matrix of L + J Nov. 21, 2012 10 / 16
Properties of Gram matrix K = ( L + J ) − 1 . Theorem 3 (Seto-S-Taniguchi) . Let T be a tree with n -vertices. Let i, j be adjacent vertices in T . Then ( ⌈ n 2 ⌉− 1)( ⌊ n 2 ⌋− 1) K i,j ≥ − with equality if and only if T is a double star n 2 T ∗ 2 ⌋− 1 and i, j are the two internal vertices. ⌈ n 2 ⌉− 1 , ⌊ n . Sketch of the proof: 1. Let T 1 (resp. T 2 ) denote the subtree of T − { i, j } containing i (resp. j ) with n 1 -vertices (resp. n 2 ). Then K ( T ) i,j = n 2 n 2 K ( T 1 ) i,i + n 2 n 2 K ( T 2 ) j,j − n 1 n 2 +1 . 1 2 n 2 Sho Suda (International Christian Univ.) On the inverse matrix of L + J Nov. 21, 2012 11 / 16
Related work on doubly stochastic graph matrices ▶ Ω = ( L + I ) − 1 is called a doubly stochastic graph matrix. . Proposition (Merris 1997) . Ω is a doubly stochastic matrix, namely all entries are nonnegative and each row and column sum is 1 . Moreover all entries are positive if and only if G is connected. . Sho Suda (International Christian Univ.) On the inverse matrix of L + J Nov. 21, 2012 12 / 16
Related work on doubly stochastic graph matrices ▶ Ω = ( L + I ) − 1 is called a doubly stochastic graph matrix. . Theorem (X.D. Zhang 2011) . Let T be a tree with n vertices, Ω a doubly stochastic matrix. Then Ω ≤ Ω( P n ) with right equality if and only if G is the path. . . Theorem (X.D. Zhang -J.X. Wu 2005) . Let T be a tree with n vertices, Ω a doubly stochastic graph matrix of T . 1 Then Ω( P n ) ≤ Ω ≤ 2( n +1) with left equality if and only if T is the path and with right equality if and only if T is the star. . . Theorem (X.D. Zhang 2005) . Let Ω be a doubly stochastic matrix. If i and j are adjacent, Then 4 Ω i,j ≥ 2 ⌋ +3) with equality if and only if T is a double star ( ⌈ n 2 ⌉ +3)( ⌊ n T ∗ 2 ⌋− 1 and i, j are the two internal vertices. ⌈ n 2 ⌉− 1 , ⌊ n . Sho Suda (International Christian Univ.) On the inverse matrix of L + J Nov. 21, 2012 13 / 16
Further problems . Theorem (Merris 1997) . Let G a graph with n vertices. Let F be the set of all spanning forests of G and F ( i, j ) be the set of spanning forests of G with both i, j belonging to the same component. For F ∈ F , γ ( F ) denotes the product of the number of connected component of F and γ i ( F ) denotes the product of the number of connected component of F that do not contain i . Then ∑ F ∈F ( i,j ) γ i ( F ) Ω i,j = . ∑ F ∈F γ ( F ) . ▶ What is an analogue of the Theorem above for the case of K ? Sho Suda (International Christian Univ.) On the inverse matrix of L + J Nov. 21, 2012 14 / 16
Further problems . Theorem (Merris 1997) . Let G a graph with n vertices and doubly stochastic graph matrix Ω . If 4 Ω i,j < n 2 +4 n , then i and j are not adjacent. . ▶ What is an analogue of the Theorem above for the case of K ? Sho Suda (International Christian Univ.) On the inverse matrix of L + J Nov. 21, 2012 15 / 16
Further problems The bounds on the entries of Ω and K look very similar, however methods are completely different. Is there a unifying way to consider both Ω and K simultaneously? How about ( L + xI + yJ ) − 1 for nonnegative real numbers x and y ? Thank you for your attention! Sho Suda (International Christian Univ.) On the inverse matrix of L + J Nov. 21, 2012 16 / 16
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