We generalize three classical characterizations of line graphs to signed and gain graphs: the Krausz's characterization, van Rooij Wilf's characterization Beineke's characterization. In particular, we present a list forbidden subgraphs characterizing class gain-line graphs. case graph whose underlying is graph, this consists exactly four Under same hypothesis, prove that if only its eigenvalues...

A signed k-partite graph (signed multipartite graph) is a k-partite graph in which each edge is assigned a positive or a negative sign. If G(V1, V2, · · · , Vk) is a signed k-partite graph with Vi = {vi1, vi2, · · · , vini}, 1 ≤ i ≤ k, the signed degree of vij is sdeg(vij) = dij = d + ij − d − ij , where 1 ≤ i ≤ k, 1 ≤ j ≤ ni and d + ij(d − ij) is the number of positive (negative) edges inciden...

A signed graph is a pair $(G, \sigma)$, where $G$ (loops and multi edges allowed) $\sigma: E(G) \to \{+, -\}$ signature which assigns to each edge of sign. Various notions coloring graphs have been studied. In this paper, we extend circular graphs. Given \sigma)$ with no positive loop, $r$-coloring an assignment $\psi$ points circle circumference $r$ the vertices such that for every $e=uv$ $G$,...

Abstract There are many concepts of signed graph coloring which defined by assigning colors to the vertices graphs. These usually differ in number self-inverse used. We introduce a unifying concept for this kind elements from symmetric sets In first part paper, we study colorings with where is fixed. prove Brooks’-type theorem and upper bounds corresponding chromatic numbers terms underlying gr...