Eigenvalues and clique partitions of graphs

A clique partition ε of graph G is a set of cliques such that each edge of G belongs to exactly one clique, and the total size of ε is the sum of cardinalities of all elements in ε. The ε-degree of a vertex u is the number of cliques in ε containing u. We say that ε is a k-restricted clique partitio...

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Bibliographic Details
Published inAdvances in applied mathematics Vol. 129; p. 102220
Main Authors Zhou, Jiang, Bu, Changjiang
Format Journal Article
LanguageEnglish
Published Elsevier Inc 01.08.2021
Subjects
Clique partition
Clique partition number
Graph spectra
Restricted clique partition number
Restricted clique partition number
Graph spectra
Clique partition number
05C70
05C50
Clique partition
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Summary:A clique partition ε of graph G is a set of cliques such that each edge of G belongs to exactly one clique, and the total size of ε is the sum of cardinalities of all elements in ε. The ε-degree of a vertex u is the number of cliques in ε containing u. We say that ε is a k-restricted clique partition if each vertex has ε-degree at least k. The (k-restricted) clique partition number of G is the smallest cardinality of a (k-restricted) clique partition of G. In this paper, we obtain eigenvalue bounds for ε-degrees, clique partition number and restricted clique partition number of a graph. As applications, we derive the De Bruijn-Erdős Theorem from our eigenvalue bounds, obtain accurate estimation of the 2-restricted clique partition number of line graphs, and give spectral lower bounds for the minimum total size of clique partitions of a graph.
ISSN:0196-8858
1090-2074
DOI:10.1016/j.aam.2021.102220