We conjecture a new lower bound on the algebraic connectivity of graph that involves number vertices high eccentricity in graph. prove this implies strengthening Laplacian Spread Conjecture. discuss further conjectures, also Conjecture, include for simple graphs and weighted graphs.

Many important graph-theoretic notions can be encoded as counting graph homomorphism problems, such partition functions in statistical physics, particular independent sets and colourings. In this article, we study the complexity of # p H OMS T O , problem homomorphisms from an input to a modulo prime number . Dyer Greenhill proved dichotomy stating that tractability non-modular depends on struc...

The Markoff conjecture states that given a positive integer c, there is at most one triple (a, b, c) of positive integers with a ≤ b ≤ c that satisfies the equation a2 + b2 + c2 = 3abc. The conjecture is known to be true when c is a prime power or two times a prime power. We present an elementary proof of this result. We also show that if in the class group of forms of discriminant d = 9c2 − 4,...

The generating graph \(\Gamma(G)\) of a finite group \(G\) has vertex set the non-identity elements \(G\), with two adjacent exactly when they generate \(G\). A coclique in is an empty induced subgraph, so subset such that no pair maximal if it contained larger coclique. It easy to see subgroup form \(\Gamma(G)\), but this need not be maximal. In paper we determine intransitive subgroups \(\mat...

‎let $g$ be a graph and $chi^{prime}_{aa}(g)$ denotes the minimum number of colors required for an‎ ‎acyclic edge coloring of $g$ in which no two adjacent vertices are incident to edges colored with the same set of colors‎. ‎we prove a general bound for $chi^{prime}_{aa}(gsquare h)$ for any two graphs $g$ and $h$‎. ‎we also determine‎ ‎exact value of this parameter for the cartesian product of ...