let $r$ be a commutative ring with identity. let $g(r)$ denote the maximal graph associated to $r$, i.e., $g(r)$ is a graph with vertices as the elements of $r$, where two distinct vertices $a$ and $b$ are adjacent if and only if there is a maximal ideal of $r$ containing both. let $gamma(r)$ denote the restriction of $g(r)$ to non-unit elements of $r$. in this paper we study the various graphi...

There are six classes of endomorphisms for a graph. The sets these form chain under the inclusion sets. In order to systematically study endomorphisms, Böttcher and Knauer defined concepts endomorphism spectrum type graph in 1992. this paper, based on property structure monoids graphs, double-edge fan graphs described. particular, we give spectra types graphs.

In this paper, we present the various upper and lower bounds for the product version of reciprocal degree distance in terms of other graph inavriants. Finally, we obtain the upper bounds for the product version of reciprocal degree distance of the composition, Cartesian product and double of a graph in terms of other graph invariants including the Harary index and Zagreb indices. .

LetD = (V (D), A(D)) be a digraph. The competition graph ofD, is the graphwith vertex set V (D) and edge set {uv ∈ ( V (D) 2 ) : ∃w ∈ V (D), uw, vw ∈ A(D)}. The double competition graph of D, is the graph with vertex set V (D) and edge set {uv ∈ ( V (D) 2 )

Let G be a finite non-abelian group and denote by Z(G) its center. The non-commuting graph of on transversal the center is whose vertices are non-central elements in two x y adjacent whenever xy=yx. In this work, we classify groups centeris double-toroidal or 1-planar.

We show that any graph that contains k edge-disjoint double rays for any k ∈ N contains also infinitely many edge-disjoint double rays. This was conjectured by Andreae in 1981.

The complete double vertex graph $M_2(G)$ of $G$ is defined as the whose vertices are $2$-multisubsets $V(G)$, and two such adjacent in if their symmetric difference (as multisets) a pair $G$. In this paper we exhibit an infinite family graphs (containing Hamiltonian non-Hamiltonian graphs) for which Hamiltonian.