Eigenvalues and clique partitions of graphs

A clique partition epsilon of graph G is a set of cliques such that each edge of G belongs to exactly one clique, and the total size of epsilon is the sum of cardinalities of all elements in epsilon. The epsilon-degree of a vertex u is the number of cliques in epsilon containing u. We say that epsilon is a k-restricted clique partition if each vertex has epsilon-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 epsilon-degrees, clique partition number and restricted clique partition number of a graph. As applications, we derive the De Bruijn-Erdos 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. (C) 2021 Elsevier Inc. All rights reserved.