Search

For a bipartite graph, the complete adjacency matrix is not necessary to display its adjacency information. In 1985, Godsil used a smaller size matrix to represent this, known as the bipartite adjacency matrix. Recently, the bipartite distance matrix of a tree with perfect matching was introduced as a concept similar to the bipartite adjacency matrix. It has been observed that these matrices are nonsingular, and a combinatorial formula for their determinants has been derived. In this article, we provide a combinatorial description of the inverse of the bipartite distance matrix and establish identities similar to some well-known identities. The study leads us to an unexpected generalization of the usual Laplacian matrix of a tree. This generalized Laplacian matrix, which we call the bipartite Laplacian matrix, is usually not symmetric, but it shares many properties with the usual Laplacian matrix. In addition, we study some of the fundamental properties of the bipartite Laplacian matrix and compare them with those of the usual Laplacian matrix.

In this article, we show that the rank of the 2-Steiner distance matrix of a caterpillar graph having NN vertices and pp pendant veritices is 2N−p−12N-p-1.

Graphs G and H are called cospectral if they have the same characteristic polynomial. If eigenvalues are integral, then corresponding graphs are called integral graph. In this article we introduce a construction to produce pairs of cospectral integral regular graphs. Generalizing the construction of G4(a, b) and G5(a, b) due to Wang and Sun, we define graphs 𝒢4(G,H) and 𝒢5(G,H) and show that they are cospectral integral regular when G is an integral q-regular graph of order m and H is an integral q-regular graph of order (b − 2)m for some integer b ≥ 3.

By a hollow symmetric matrix we mean a symmetric matrix with zero diagonal elements. The notion contains those of predistance matrix and Euclidean distance matrix as its special cases. By a centered symmetric matrix we mean a symmetric matrix with zero row (and hence column) sums. There is a one-toone correspondence between the classes of hollow symmetric matrices and centered symmetric matrices, and thus with any hollow symmetric matrix D we may associate a centered symmetric matrix B, and vice versa. This correspondence extends a similar correspondence between Euclidean distance matrices and positive semidefinite matrices with zero row and column sums.We show that if B has rank r, then the corresponding D must have rank r, r + 1 or r + 2. We give a complete characterization of the three cases.We obtain formulas for the Moore-Penrose inverse D+ in terms of B+, extending formulas obtained in Kurata and Bapat (Linear Algebra and Its Applications, 2015). If D is the distance matrix of a weighted tree with the sum of the weights being zero, then B+ turns out to be the Laplacian of the tree, and the formula for D+ extends a well-known formula due to Graham and Lovász for the inverse of the distance matrix of a tree.

Let T be a tree with vertex set {1,...,n} such that each edge is assigned a nonzero weight. The squared distance matrix of T, denoted by Δ, is the n×n matrix with (i,j)-element d(i,j)2, where d(i,j) is the sum of the weights of the edges on the (ij)-path. We obtain a formula for the determinant of Δ. A formula for Δ-1 is also obtained, under certain conditions. The results generalize known formulas for the unweighted case. [ABSTRACT FROM AUTHOR]

Let G be a graph (directed or undirected) having k number of blocks B1, B2, . . . , Bk. A β-partition of G is a partition consists of k vertex-disjoint subgraph (B̂1, B̂1, . . . , B̂k) such that B̂i is an induced subgraph of Bi for i = 1, 2, . . . , k. The terms ∏ki=1 det(B̂i), ∏ki=1 per(B̂i) represent the det-summands and the persummands, respectively, corresponding to the β-partition (B̂1, B̂1, . . . , B̂k). The determinant (permanent) of a graph having no loops on its cut-vertices is equal to the summation of the det-summands (per-summands), corresponding to all possible β-partitions. In this paper, we calculate the determinant and the permanent of classes of graphs such as block graph, block graph with negatives cliques, signed unicyclic graph, mixed complete graph, negative mixed complete graph, and star mixed block graphs. [ABSTRACT FROM AUTHOR]

Graphs G and H are called cospectral if they have the same characteristic polynomial, equivalently, if they have the same eigenvalues considering multiplicities. In this article we introduce a construction to produce pairs of cospectral regular graphs. We also investigate conditions under which the graphs are integral. [ABSTRACT FROM AUTHOR]