For an undirected graph G, a zero-sum flow is an assignment of non-zero real numbers to the edges, such that the sum of the values of all edges incident with each vertex is zero. It has been conjectured that if a graph G has a zero-sum flow, then it has a zero-sum 6-flow. We prove this conjecture and Bouchet’s Conjecture for bidirected graphs are equivalent. Among other results it is shown that...

Let $G$ be a bipartite graph. In this paper we consider the two kind of location problems namely $p$-center and $p$-median problems on bipartite graphs. The $p$-center and $p$-median problems asks to find a subset of vertices of cardinality $p$, so that respectively the maximum and sum of the distances from this set to all other vertices in $G$ is minimized. For each case we present some proper...

The spectral graph theory explores connections between combinatorial features of graphs and algebraic properties associated matrices. neighborhood inverse sum indeg (NI) index was recently proposed explored to be a significant molecular descriptor. Our aim is investigate the NI from standpoint, for which suitable matrix proposed. symmetric since it generated edge connection information undirect...

The 1–2–3 Conjecture, raised by Karoński, Łuczak and Thomason, states that almost every graph G admits a proper 3-labelling, i.e., labelling of the edges with 1, 2, 3 such no two adjacent vertices are incident to same sum labels. Another interpretation this conjecture, may be attributed Chartrand et al., is can turned into locally irregular multigraph M, having degree, replacing each its at mos...