Let G be a non-trivial group, S ⊆ G \ {1} and S = S−1 := {s−1 | s ∈ S}. The Cayley graph of G denoted by Γ(S : G) is a graph with vertex set G and two vertices a and b are adjacent if ab−1 ∈ S. A graph is called integral, if its adjacency eigenvalues are integers. In this paper we determine all connected cubic integral Cayley graphs. We also introduce some infinite families of connected integra...

let $r$ be a commutative ring with identity and $m$ an $r$-module. in this paper, we associate a graph to $m$, say ${gamma}({}_{r}m)$, such that when $m=r$, ${gamma}({}_{r}m)$ coincide with the zero-divisor graph of $r$. many well-known results by d.f. anderson and p.s. livingston have been generalized for ${gamma}({}_{r}m)$. we show that ${gamma}({}_{r}m)$ is connected with ${diam}({gamma}({}_...

for a simple connected graph $g$ with $n$-vertices having laplacian eigenvalues‎ ‎$mu_1$‎, ‎$mu_2$‎, ‎$dots$‎, ‎$mu_{n-1}$‎, ‎$mu_n=0$‎, ‎and signless laplacian eigenvalues $q_1‎, ‎q_2,dots‎, ‎q_n$‎, ‎the laplacian-energy-like invariant($lel$) and the incidence energy ($ie$) of a graph $g$ are respectively defined as $lel(g)=sum_{i=1}^{n-1}sqrt{mu_i}$ and $ie(g)=sum_{i=1}^{n}sqrt{q_i}$‎. ‎in th...

Let k be a non-negative integer. A branch vertex of a tree is a vertex of degree at least three. We show two sufficient conditions for a connected clawfree graph to have a spanning tree with a bounded number of branch vertices: (i) A connected claw-free graph has a spanning tree with at most k branch vertices if its independence number is at most 2k + 2. (ii) A connected clawfree graph of order...

the non commuting graph of a non-abelian finite group $g$ is defined as follows: its vertex set is $g-z(g)$ and two distinct vertices $x$ and $y$ are joined by an edge if and only if the commutator of $x$ and $y$ is not the identity. in this paper we prove some new results about this graph. in particular we will give a new proof of theorem 3.24 of [2]. we also prove that if $g_1$, $g_2$, ..., $...

In this note we show that the edge-connectivity λ(G × H) of the direct product of graphs G and H is bounded below by min{λ(G)|E(H)|, λ(H)|E(G)|, δ(G × H)} and above by min{2λ(G)|E(H)|, 2λ(H)|E(G)|, δ(G×H)} except in some special cases when G is a relatively small bipartite graph, or both graphs are bipartite. Several upper bounds on the vertex-connectivity of the direct product of graphs are al...

In 1966, Gallai asked whether every connected graph has a vertex that is common to all its longest paths. The answer to this question is negative. We prove that the answer is positive for outerplanar graphs. Another related question was raised in 1995 at the British Combinatorial Conference: Do any three longest paths in a connected graph have a vertex in common? We prove that, in a connected g...

general formulas are obtained for the vertex padmakar-ivan index (piv) of tetrathiafulvalene(ttf) dendrimer, whereby ttf units we are employed as branching centers. the piv index isa wiener-szeged-like index developed very recently. this topological index is defined as thesummation of all sums of nu(e) and nv(e), over all edges of connected graph g.

let g be a simple connected graph. the generalized polarity wiener index ofg is defined as the number of unordered pairs of vertices of g whosedistance is k. some formulas are obtained for computing the generalizedpolarity wiener index of the cartesian product and the tensor product ofgraphs in this article.