‎‎let r be a non-commutative ring with unity‎. ‎the commuting graph‎ ‎of $r$ denoted by $gamma(r)$‎, ‎is a graph with a vertex set‎ ‎$rsetminus z(r)$ and two vertices $a$ and $b$ are adjacent if and only if‎ ‎$ab=ba$‎. ‎in this paper‎, ‎we investigate non-commutative rings with unity of order $p^n$ where $p$ is prime and $n in lbrace 4,5 rbrace$‎. it is shown that‎, ‎$gamma(r)$ is the disjoint ...

in recent decades, phytoremediation is one of the methods which has been a lot of attention. in this process by using of green plants can be removing or eliminate pollutants from water, soil and sediments. therefore, to achievement the objective, this experiment was done in pot culture using completely randomized design with different levels of cadmium concentrations (cdcl2.h2o) including 0, 5,...

let $n$ be any positive integer, the friendship graph $f_n$ consists of $n$ edge-disjoint triangles that all of them meeting in one vertex. a graph $g$ is called cospectral with a graph $h$ if their adjacency matrices have the same eigenvalues. recently in href{http://arxiv.org/pdf/1310.6529v1.pdf}{http://arxiv.org/pdf/1310.6529v1.pdf} it is proved that if $g$ is any graph cospectral with $f_n$...

This paper gives lower bounds on the spectral radius of vertex-transitive graphs, based on the number of “prime cycles” at a vertex. The bounds are obtained by constructing circuits in the graph that resemble “cactus trees”, and enumerating them. Counting these circuits gives a coefficient-wise underestimation of the Green function of the graph, and hence an underestimation of its spectral radi...

This paper concerns random walks on periodic graphs embedded in the d-dimensional Euclidian space Rd and obtains asymptotic expansions of the Green functions of them up to the second order term, which, expressed fairly explicitly, are easily computable for many examples. The result is used to derive an asymptotic form of the hitting distribution of a hyperplane of codimension one, which involve...

In 1977 a five-part conjecture was made about a family of groups related to trivalent graphs and subsequently two parts of the conjecture were proved. The conjecture completely determines all finite members of the family. Here we complete the proof of the conjecture by giving proofs for the remaining three parts. Dedicated to Charles Leedham-Green on the occasion of his 65th birthday

Regular boundary value problems on a distance-regular graph associated with Schrödinger operators are analyzed. These problems include the cases in which the boundary has one or two vertices. In each case, the Green matrices are given in terms of two families of orthogonal polynomials, one of them corresponding with the distance polynomials of the distance-regular graphs.