Various models to quantify the reliability of a network have been studied where certain components graph may fail at random and probability that remaining is connected proxy for reliability. In this work we introduce strengthening one these by considering not just but also cop-win. A cop-win if cop can guarantee capture fleeing robber in well-studied pursuit-evasion game Cops Robber. More preci...

Let Sz(G), Sz(G) and W (G) be the Szeged index, revised Szeged index and Wiener index of a graph G. In this paper, the graphs with the fourth, fifth, sixth and seventh largest Wiener indices among all unicyclic graphs of order n > 10 are characterized; and the graphs with the first, second, third, and fourth largest Wiener indices among all bicyclic graphs are identified. Based on these results...

The conjecture Σuv V(G) dG(u)2 / n(G) ≤ Σuvv E(G) dG(u)dG(v) / m(G) that compares normalized Zagreb indices attracted recently a lot of attention1-9. In this paper we analyze analogous statement in which degree dG(u) of vertex u is replaced by its eccentricity δG(u) in which way we define novel first and second Zagreb eccentricity indices. We show that Σuv V(G) εG(u)2 / n(G) ≥ Σuvv E(G) εG(u)εG...

Let G be a connected graph; the edge Mostar index Moe(G) of is defined as Moe(G)=∑e=uv∈E(G)|mu(e)−mv(e)|, where mu(e) and mv(e) denote number edges in that are closer to vertex u than v u, respectively. In this paper, we determine upper bound for all bicyclic graphs identify extremal achieve bound.

Bicyclic Mixed Triple Systems by Benkam Benedict Bobga In the study of triple systems, one question faced is that of finding for what order a decomposition exists. We state and prove a necessary and sufficient condition for the existence of a bicyclic mixed triple system based on the three possible partial orientations of the 3-cycle with twice as many arcs as edges. We also explore the existen...

The study of the maximum and minimal characteristics graphs is focus significant field mathematics known as extreme graph theory. Finding biggest or smallest that meet specified criteria main goal this discipline. There are several applications extremal theory in various fields, including computer science, physics, chemistry. Some important include: Computer networking, social chemistry physics...