The \(m\)-Clique Load and the \(m\)-Clique Sequence of Graphs

Abstract

This paper introduces the concepts of the \(m\)-clique load, the \(m\)-clique sequence and the \(m\)-clique density of graphs. The number of distinct maximum cliques over all maximal cliques is called the \(m\)-clique load of \(G\) and denoted, \(\diamond(G)\). The \(m\)-clique sequence denoted, \(\diamond\)-sequence of a graph \(G\) with \(\epsilon(G)\ge 1\) is the sequence with entries representing the number of maximal cliques of same order found in \(G\), in descending order. A finite sequence of positive integers each indexed with a distinct positive integer subscript which is \(c\)-graphical, is characterised. The \(m\)-clique density of a graph \(G\) denoted, \(p_{c_i}(G)\) is the probability of uniformly at random, choosing a maximal clique \(K_{c_i}\), \(1\le c_i\le \nu(G)\). Introductory results for certain graph classes and power graphs of balanced caterpillars, \(C^{\cal L}_{P_n}\) are also presented.