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).

A projective-planar signed graph has no two vertex-disjoint negative circles. We prove that every signed graph with no two vertex-disjoint negative circles and no balancing vertex is obtained by taking a projective-planar signed graph or a copy of −K5 and then taking 1, 2, and 3-sums with balanced signed graphs.

It is well-known that the Jones polynomial of an alternating knot is closely related to the Tutte polynomial of a special graph obtained from a regular projection of the knot. Relying on the results of Bollobás and Riordan, we introduce a generalization of Kauffman’s Tutte polynomial of signed graphs for which describing the effect of taking a signed tensor product of signed graphs is very simp...

We study the application of spectral clustering, prediction and visualization methods to graphs with negatively weighted edges. We show that several characteristic matrices of graphs can be extended to graphs with positively and negatively weighted edges, giving signed spectral clustering methods, signed graph kernels and network visualization methods that apply to signed graphs. In particular,...

The nullity of a graph is the multiplicity of the eigenvalues zero in its spectrum. A signed graph is a graph with a sign attached to each of its edges. In this paper, we obtain the coefficient theorem of the characteristic polynomial of a signed graph, give two formulae on the nullity of signed graphs with cut-points. As applications of the above results, we investigate the nullity of the bicy...