Odd-Even Clique Equality in the Coprime Graph of the Generalised Quaternion Group

Abstract

Properties of various groups are investigated by associating them with graphs. The quaternion group, which is non-abelian, is one of the most explored groups. Several studies have been carried out on this group by associating it with various graphs, such as commuting graph, centre graph, conjugate graph and coprime graph. Meanwhile, the clique polynomial is determined to investigate the properties and relations of the cliques of graphs. The objectives of this paper are to determine and investigate the general forms of the clique polynomial of the coprime graph of the generalised quaternion group by using the prime factorisations of the order of the group. The algorithm for the coefficients of the clique polynomials is discussed using combinatorial methods and the partitioning of the sets of vertices. The results show that the total number of odd-sized cliques of the coprime graph of the generalised quaternion group is equal to the total number of even-sized cliques.