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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.

In this paper, several existing results related to secondary transpose are critically reviewed and a result analogous to spectral decomposition theorem is obtained for a real secondary symmetric matrix. Noting that Moore–Penrose inverse with reference to secondary transpose involution, namely s -g inverse, need not always exist, we explore a few necessary sufficient conditions for the existence of such Moore–Penrose inverse. Further, we provide expressions and determinantal formula to compute the same. [ABSTRACT FROM AUTHOR]

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.

Let G be a connected graph with V (G) = { 1 , ... , n }. Then the resistance distance between any two vertices i and j is given by r i j := l i i † + l j j † − 2 l i j † , where l i j † is the (i , j) th entry of the Moore-Penrose inverse of the Laplacian matrix of G. For the resistance matrix R := [ r i j ] , there is an elegant formula to compute the inverse of R. This says that R − 1 = − 1 2 L + 1 τ ′ R τ τ τ ′ , where τ := (τ 1 , ... , τ n) ′ and τ i := 2 − ∑ j ∼ i r i j i = 1 , ... , n. A far reaching generalization of this result that gives an inverse formula for a generalized resistance matrix of a strongly connected and matrix weighted balanced directed graph is obtained in this paper. When the weights are scalars, it is shown that the generalized resistance is a non-negative real number. We also obtain a perturbation result involving resistance matrices of connected graphs and Laplacians of digraphs. [ABSTRACT FROM AUTHOR]

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.

The distance matrix of a simple connected graph G is D (G) = (d i j) , where d i j is the distance between the vertices i and j in G. We consider a weighted tree T on n vertices with edge weights are square matrices of the same size. The distance d i j between the vertices i and j is the sum of the weight matrices of the edges in the unique path from i to j. In this article, we establish a characterization for the trees in terms of the rank of (matrix) weighted Laplacian matrix associated with it. We present a necessary and sufficient condition for the distance matrix D, with matrix weights, to be invertible and the formula for the inverse of D, if it exists. Then we study some of the properties of the distance matrices of matrix weighted trees in connection with the Laplacian matrices, incidence matrices, and g-inverses. Finally, we derive an interlacing inequality for the eigenvalues of distance and Laplacian matrices for the case of positive definite matrix weights. [ABSTRACT FROM AUTHOR]

The resistance matrix of a simple connected graph G is denoted by R G or simply by R and is defined by R G = (r i j) , where r i j is the resistance distance between the vertices i and j in G. In this paper, we consider distance matrix of a weighted tree and the resistance matrix of any weighted graph, where the weights are nonzero scalars. We obtain the identities for minors of the Laplacian, resistance and distance matrices, which are independent of the nonsingularity of resistance and distance matrices. While finding these we obtain the necessary and sufficient condition for the resistance matrix to be singular and the rank of it. Finally, we obtain the Moore–Penrose inverse of R, when it is singular. [ABSTRACT FROM AUTHOR]

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]