On -commuting graph of matrix algebra.
On composition of signed graphs
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,...
On conditional independence and log-convexity
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.
On connected graphs with maximal index.
On determining minimal spectrally arbitrary patterns.
On digraphs with non-derogatory adjacency matrix.
On eigenvectors of mixed graphs with exactly one nonsingular cycle
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...
On formal products and the Seidel spectrum of graphs.
On Graphs Whose Energy Does Not Exceed 4
On graphs whose reduced energy does not exceed 3.
On Graphs Whose Second Largest Eigenvalue Does Not Exceed 1
On graphs whose second least eigenvalue is at least
On graphs whose spectral spread does not exceed 4.
On graphs with cyclic defect or excess.
On graphs with the largest Laplacian index
Let be a connected simple graph on vertices. The Laplacian index of , namely, the greatest Laplacian eigenvalue of , is well known to be bounded above by . In this paper, we give structural characterizations for graphs with the largest Laplacian index . Regular graphs, Hamiltonian graphs and planar graphs with the largest Laplacian index are investigated. We present a necessary and sufficient condition on and for the existence of a -regular graph of order with the largest Laplacian...
On integer sequences associated with the cyclic and complete graphs.
On -walk-regular graphs.
On Laplacian eigenvalues of connected graphs
Let be an undirected connected graph with , , vertices and edges with Laplacian eigenvalues . Denote by , , , the sum of arbitrary Laplacian eigenvalues, with and . Lower bounds of graph invariants and are obtained. Some known inequalities follow as a special case.
On minimal energies of trees with given diameter.
On minimal rank over finite fields.