let $g$ be a finite group. we construct the prime graph of $ g $,which is denoted by $ gamma(g) $ as follows: the vertex set of thisgraph is the prime divisors of $ |g| $ and two distinct vertices $ p$ and $ q $ are joined by an edge if and only if $ g $ contains anelement of order $ pq $.in this paper, we determine finite groups $ g $ with $ gamma(g) =gamma(l_3(q)) $, $2 leq q < 100 $ and prov...

Let $G$ be a finite group. We construct the prime graph of $ G $,which is denoted by $ Gamma(G) $ as follows: the vertex set of thisgraph is the prime divisors of $ |G| $ and two distinct vertices $ p$ and $ q $ are joined by an edge if and only if $ G $ contains anelement of order $ pq $.In this paper, we determine finite groups $ G $ with $ Gamma(G) =Gamma(L_3(q)) $, $2 leq q < 100 $ and prov...

in this paper we will prove that the simple group g2(q) where 2 < q = 1(mod3)is recognizable by the set of its order components, also other word we prove that if g is anite group with oc(g) = oc(g2(q)), then g is isomorphic to g2(q).

An old conjecture of Ringel states that every tree with m edges decomposes the complete graph K2m+1. The best lower bound for the order of a complete graph decomposed by a given tree with m edge is O(m). We show that asymptotically almost surely a random tree with m edges and p = 2m + 1 a prime decomposes K2m+1(r) for every r ≥ 2, the graph obtained from the complete graph K2m+1 by replacing ea...

Let p be a prime and n a positive integer. In [J. Austral. Math. Soc. 81 (2006), 153–164], Feng and Kwak showed that if p > 5 then every connected cubic symmetric graph of order 2p is a Cayley graph. Clearly, this is not true for p = 5 because the Petersen graph is non-Cayley. But they conjectured that this is true for p = 3. This conjecture is confirmed in this paper. Also, for the case when p...

Suppose $X$ is a simple graph. The $X-$join $Gamma$ of a set ofcomplete or empty graphs ${X_x }_{x in V(X)}$ is a simple graph with the following vertex and edge sets:begin{eqnarray*}V(Gamma) &=& {(x,y) | x in V(X) & y inV(X_x) },\ E(Gamma) &=& {(x,y)(x^prime,y^prime) | xx^prime in E(X) or else x = x^prime & yy^prime in E(X_x)}.end{eqnarray*}The $X-$join graph $Gamma$ is said to be re...

Mader conjectured that for any tree T $T$ of order m $m$ , every k $k$ -connected graph G $G$ with minimum degree at least ⌊ 3 2 ⌋ + − 1 $\lfloor \frac{3k}{2}\rfloor +m-1$ contains a subtree ′ ≅ $T^{\prime} \cong T$ such V ( ) $G-V(T^{\prime} )$ is -connected. In this article, we give characterization subgraph to contain an embedding specified avoiding some vertex. As corollary, confirm Mader's...

let $gamma(s_n)$ be the minimum number of proper subgroups‎ ‎$h_i, i=1‎, ‎dots‎, ‎l $ of the symmetric group $s_n$ such that each element in $s_n$‎ ‎lies in some conjugate of one of the $h_i.$ in this paper we‎ ‎conjecture that $$gamma(s_n)=frac{n}{2}left(1-frac{1}{p_1}right)‎ ‎left(1-frac{1}{p_2}right)+2,$$ where $p_1,p_2$ are the two smallest primes‎ ‎in the factorization of $ninmathbb{n}$ an...

A multicone graph is defined to be join of a clique and a regular graph. A graph $ G $ is cospectral with graph $ H $ if their adjacency matrices have the same eigenvalues. A graph $ G $ is said to be determined by its spectrum or DS for short, if for any graph $ H $ with $ Spec(G)=Spec(H)$, we conclude that $ G $ is isomorphic to $ H $. In this paper, we present new classes of multicone graphs...

Jean-Claude Bermond, Eric Darrot, Olivier Delmas, Stéphane Perennes* Thème 1 — Réseaux et systèmes Projet SLOOP Rapport de recherche n ̊???? — Juillet 1996 — 25 pages Abstract: in this paper, we prove that the wrapped Butterfly digraph ~ WBF(d; n) of degree d and dimensionn contains at least d 1 arc-disjoint Hamilton circuits, answering a conjecture of D. Barth. We also conjecture that ~ WBF(d; ...