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Spectral Applications of Vertex-Clique Incidence Matrices Associated with a Graph
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: Received: 19 July 2023 / Approved: 19 July 2023 / Online: 19 July 2023 (11:47:31 CEST)
A peer-reviewed article of this Preprint also exists.
Fallat, S.; Mojallal, S.A. Spectral Applications of Vertex-Clique Incidence Matrices Associated with a Graph. Mathematics 2023, 11, 3595. Fallat, S.; Mojallal, S.A. Spectral Applications of Vertex-Clique Incidence Matrices Associated with a Graph. Mathematics 2023, 11, 3595.
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 $q(G)=2$.
Keywords
clique partition; edge clique cover; vertex-clique incidence matrix; eigenvalues of graphs; graph energy; minimum number of distinct eigenvalues
Subject
Computer Science and Mathematics, Discrete Mathematics and Combinatorics
Copyright: This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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