Some Results on Spectra and Energy of Edge-Connectivity Matrices of Graphs

The spectrum of a matrix refers to the multiset of its eigenvalues, and its spectral radius refers to the maximum absolute value of its eigenvalues. The energy of a matrix is defined as the sum of the absolute values of all its eigenvalues. For a connected graph [Formula: see text] with vertex set [Formula: see text], the (vertex-) edge-connectivity matrix of [Formula: see text] is a square matrix of order [Formula: see text] whose corresponding entry in row [Formula: see text] and column [Formula: see text] is the maximum number of (internally vertex-) edge-disjoint paths connecting [Formula: see text] and [Formula: see text] for [Formula: see text]; otherwise, the corresponding entry is 0. The spectra and energy of vertex-connectivity matrices of graphs have been considered extensively in the existing literature. Recently, Akbari et al. [On edge-path eigenvalues of graphs, Linear Multilinear Algebra, 70(15)(2022) 2998–3008.] further introduced the edge-connectivity matrix of a graph, and presented some properties of edge-connectivity matrices with respect to the edge-connectivity of several graph classes. In this paper, we determine the spectra of edge-connectivity matrices for several special graph classes, and obtain some properties of the edge-connectivity matrices of graphs and upper and lower bounds on the spectral radii of the edge-connectivity matrices for several graph classes. Moreover, we establish upper and lower bounds on the energy of edge-connectivity matrices of graphs, and further obtain the exact values of the energy of the edge-connectivity matrices for bicyclic graphs of Type [Formula: see text] and Cartesian products of several special graph classes.