let $g$ be a group. the order graph of $g$ is the (undirected)graph $gamma(g)$,those whose vertices are non-trivial subgroups of $g$ and two distinctvertices $h$ and $k$ are adjacent if and only if either$o(h)|o(k)$ or $o(k)|o(h)$. in this paper, we investigate theinterplay between the group-theoretic properties of $g$ and thegraph-theoretic properties of $gamma(g)$. for a finite group$g$, we s...

By the Von Neumann regular graph of R, we mean the graph that its vertices are all elements of R such that there is an edge between vertices x,y if and only if x+y is a von Neumann regular element of R, denoted by G_Vnr (R). For a commutative ring R with unity, x in R is called Von Neumann regular if there exists x in R such that a=a2 x. We denote the set of Von Neumann regular elements by V nr...

A graph G is claw-free if no induced subgraph of it is isomorphic to the complete bipartite graph K1,3, and it is prime if |V (G)| ≥ 4 and there is no X ⊆ V (G) with 1 < |X| < |V (G)| such that every vertex of V (G) \X with a neighbour in X is adjacent to every x ∈ X. A simplicial clique in G is a non-empty clique K, such that for every k ∈ K the set of neighbours of k in V (G) \K is a clique. ...

Let G be a (p, q) graph. Let f : V (G) → {1, 2, . . . , k} be a function. For each edge uv, assign the label gcd (f(u), f(v)). f is called k-prime cordial labeling of G if ∣∣vf (i)− vf (j)∣∣ 6 1, i, j ∈ {1, 2, . . . , k} and ∣∣ef (0)− ef (1)∣∣ 6 1 where vf (x) denotes the number of vertices labeled with x, ef (1) and ef (0) respectively denote the number of edges labeled with 1 and not labeled ...

a graph is textit{symmetric}, if its automorphism group is transitive on the set of its arcs. in this paper, we  classifyall the connected cubic symmetric  graphs of order $36p$  and $36p^{2}$, for each prime $p$, of which the proof depends on the classification of finite simple groups.

A graph G(V,E) with vertex set V is said to have a prime labeling if there exist a bijection f : V (G) → {1, 2, . . . , |V |} such that for each edge xy ∈ E(G), f(x) and f(y) are relatively prime. It becomes an interesting problem to investigate the maximum possible edges that can be constructed in a graph with n vertices having prime labeling property. We interpret this problem in number theor...

a graph is called textit{circulant} if it is a cayley graph on a cyclic group, i.e. its adjacency matrix is circulant. let $d$ be a set of positive, proper divisors of the integer $n>1$. the integral circulant graph $icg_{n}(d)$ has the vertex set $mathbb{z}_{n}$ and the edge set e$(icg_{n}(d))= {{a,b}; gcd(a-b,n)in d }$. let $n=p_{1}p_{2}cdots p_{k}m$, where $p_{1},p_{2},cdots,p_{k}$ are disti...

We introduce and study the dominated edge coloring of a graph. A graph \(G\), is proper \(G\) such that each color class by at least one \(G\). The minimum number colors among all called chromatic number, denoted \(\chi_{dom}^{\prime}(G)\). obtain some properties \(\chi_{dom}^{\prime}(G)\) compute it for specific graphs. Also examine effects on \(\chi_{dom}^{\prime}(G)\), when modified operatio...

A graph G is claw-free if no induced subgraph of it is isomorphic to the complete bipartite graph K1,3, and it is prime if |V (G)| ≥ 4 and there is no X ⊆ V (G) with 1 < |X| < |V (G)| such that every vertex of V (G) \ X with a neighbour in X is adjacent to every vertex of X. In particular, if G is prime, then both G and Gc are connected. This paper has two main results. The first one is that if...