A signed graph Σ=(G,σ) is a where the function σ assigns either 1 or −1 to each edge of simple G. The adjacency matrix Σ, denoted by A(Σ), defined canonically. In recent paper, Wang et al. extended spectral bounds Hoffman and Cvetković for chromatic number graphs. They proposed an open problem related balanced clique largest eigenvalue graph. We solve strengthened version this problem. As bypro...

We introduce the ``trivariate Tutte polynomial" of a signed graph as an invariant graphs up to vertex switching that contains among its evaluations number proper colorings and nowhere-zero flows. In this, it parallels polynomial graph, which chromatic flow specializations. The tensions (for they are not simply related for graphs) is given in terms trivariate at two distinct points. Interestingl...

The index of a signed graph is the largest eigenvalue its adjacency matrix. For positive integers n and m ? 2 / 4 , we determine maximum complete graphs with vertices negative edges characterize achieving this maximum. This settles (the corrected version of) conjecture by Koledin Stani? (2017).