Spectral Applications of Vertex-Clique Incidence Matrices Associated with a Graph

Abstract

Using the notions of clique partitions and edge clique covers of graphs, we consider the corresponding incidence structures. This connection furnishes lower bounds on the negative eigenvalues and their multiplicities associated with the adjacency matrix, bounds on the incidence energy, and on the signless Laplacian energy for graphs. For the more general and well-studied set S(G) of all real symmetric matrices associated with a graph G, we apply an extended version of an incidence matrix tied to an edge clique cover to establish several classes of graphs that allow two distinct eigenvalues.

Keywords

• clique partition
• edge clique cover
• vertex-clique incidence matrix
• eigenvalues of graphs
• graph energy
• minimum number of distinct eigenvalues