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In a recent paper the authors proposed a lower bound on , where , , is an eigenvalue of a transition matrix of an ergodic Markov chain. The bound, which involved the group inverse of , was derived from a more general bound, due to Bauer, Deutsch, and Stoer, on the eigenvalues of a stochastic matrix other than its constant row sum. Here we adapt the bound to give a lower bound on the algebraic connectivity of an undirected graph, but principally consider the case of equality in the bound when...
The paper is concerned with the existence of non-negative or positive solutions to , where is the vertex-edge incidence matrix of an undirected graph. The paper gives necessary and sufficient conditions for the existence of such a solution.
A graph whose edges are labeled either as positive or negative is called a signed graph. In this article, we extend the notion of composition of (unsigned) graphs (also called lexicographic product) to signed graphs. We employ Kronecker product of matrices to express the adjacency matrix of this product of two signed graphs and hence find its eigenvalues when the second graph under composition is net-regular. A signed graph is said to be net-regular if every vertex has constant net-degree, namely,...
If conditional independence constraints define a family of positive distributions that is log-convex then this family turns out to be a Markov model over an undirected graph. This is proved for the distributions on products of finite sets and for the regular Gaussian ones. As a consequence, the assertion known as Brook factorization theorem, Hammersley–Clifford theorem or Gibbs–Markov equivalence is obtained.
For a simple connected graph of order having distance Laplacian eigenvalues , the distance Laplacian energy is defined as , where is the Wiener index of . We obtain a relationship between the Laplacian energy and the distance Laplacian energy for graphs with diameter 2. We obtain lower bounds for the distance Laplacian energy in terms of the order , the Wiener index , the independence number, the vertex connectivity number and other given parameters. We characterize the extremal graphs...
Let be a mixed graph. The eigenvalues and eigenvectors of are respectively defined to be those of its Laplacian matrix. If is a simple graph, [M. Fiedler: A property of eigenvectors of nonnegative symmetric matrices and its applications to graph theory, Czechoslovak Math. J. 25 (1975), 619–633] gave a remarkable result on the structure of the eigenvectors of corresponding to its second smallest eigenvalue (also called the algebraic connectivity of ). For being a general mixed graph with...
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