Let [see formula in PDF] be a simple graph having vertex set PDF]and edge PDF], where all are distinct prime factors and is the of units ring PDF]be signed whose underlying signature function defined as PDF]In this paper, we characterize balance some graphs derived from it such PDF]. Moreover, investigate clusterability sign-compativility

Abstract Community detection is a common task in social network analysis with applications variety of fields including medicine, criminology and business. Despite the popularity community detection, there no clear consensus on most effective methodology for signed networks. In this article, we summarize development networks evaluate current state-of-the-art techniques several real-world dataset...

A connected signed graph Ġ, where all blocks of it are positive cliques or negative (of possibly varying sizes), is called a block graph. Let A, N and D̃ be adjacency, net Laplacian distance matrices graph, respectively. In this paper the formulas for determinant were given firstly. Then inverse (resp. Moore-Penrose inverse) obtained if nonsingular singular), which sum Laplacian-like matrix at m...

This paper presents a spectral analysis of signed networks from both theoretical and practical aspects. On the theoretical aspect, we conduct theoretical study based on matrix perturbation theorem for analyzing community structures of complex signed networks and show how the negative edges affect distributions and patterns of node spectral coordinates in the spectral space. We prove and demonst...

Let G be a graph with vertex set V (G). A function f : V (G) → {−1, 1} is a signed dominating function of G if, for each vertex of G, the sum of the values of its neighbors and itself is positive. The signed domination number of a graph G, denoted γs(G), is the minimum value of ∑ v∈V (G) f(v) over all the signed dominating functions f of G. The signed reinforcement number of G, denoted Rs(G), i...

This report briefly describes the development and applications of net-sign graph theory. The current work enunciates the graph (molecule) signature of nonalternant non-benzenoid hydrocarbons with odd member of rings (non-bipartite molecular graphs) based on chemical signed graph theory. Experimental evidences and Hückel spectrum reveal that structure possessing nonbonding molecular orbital (NBM...

It is well-known that the Jones polynomial of a knot is closely related to the Tutte polynomial of a special graph obtained from a regular projection of the knot. In this paper, we study the Tutte polynomials for signed graphs. We show that if a signed graph is constructed from a simpler graph via k-thickening or k-stretching, then its Tutte polynomial can be expressed in terms of the Tutte pol...