Eigenvalues of weakly balanced signed graphs and graphs with negative cliques

TL;DR

This paper derives eigenvalues and characteristic polynomials for various signed graphs with negative cliques, including cycle, path, and complete graphs, and explores their implications in social network stability.

Contribution

It provides explicit eigenvalue formulas for classes of signed graphs with negative cliques and extends results to weakly balanced graphs by sign reversal.

Findings

Eigenvalues for cycle and path graphs with negative cliques

Eigenvalues for complete graphs with disjoint negative cliques

Eigenvalues for star block graphs with negative cliques

Abstract

In a signed graph $G$, an induced subgraph is called a negative clique if it is a complete graph and all of its edges are negative. In this paper, we give the characteristic polynomials and the eigenvalues of some signed graphs having negative cliques. This includes cycle graphs, path graphs, complete graphs with vertex-disjoint negative cliques of different orders, and star block graphs with negative cliques. Interestingly, if we reverse the signs of the edges of these graphs, we get the families of weakly balanced signed graphs, thus the eigenvalues of wide classes of weakly balanced signed graphs are also calculated. In social network theory, the eigenvalues of the signed graphs play an important role in determining their stability and developing the measures for the degree of balance.