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

TL;DR

This paper explores the spectral properties of graphs using vertex-clique incidence matrices derived from clique partitions and edge clique covers, leading to new bounds on eigenvalues, energies, and characterizations of graphs with minimal eigenvalue counts.

Contribution

It introduces the vertex-clique incidence matrix concept and generalizes spectral notions, providing new bounds and characterizations related to graph energies and eigenvalues.

Findings

New lower bounds for negative eigenvalues and inertia.

Generalizations of signless Laplacian and energy bounds.

Characterizations of graphs with minimal eigenvalue counts.

Abstract

In this paper, we demonstrate a useful interaction between the theory of clique partitions, edge clique covers of a graph, and the spectra of graphs. Using a clique partition and an edge clique cover of a graph we introduce the notion of a vertex-clique incidence matrix for a graph and produce new lower bounds for the negative eigenvalues and negative inertia of a graph. Moreover, utilizing these vertex-clique incidence matrices, we generalize several notions such as the signless Laplacian matrix, and develop bounds on the incidence energy and the signless Laplacian energy of the graph. %The tight upper bounds for the energies of a graph and its line graph are given. More generally, we also consider the set $S(G)$ of all real-valued symmetric matrices whose off-diagonal entries are nonzero precisely when the corresponding vertices of the graph are adjacent. An important parameter in this setting is $q(G)$, and is defined to be the minimum number of distinct eigenvalues over all matrices in $S(G)$. For a given graph $G$ the concept of a vertex-clique incidence matrix associated with an edge clique cover is applied to establish several classes of graphs with $q(G)=2$.