Let $G$ be a simple graph with vertex set ${v_1,v_2,ldots,v_n}$. The common neighborhood graph (congraph) of $G$, denoted by $con(G)$, is the graph with vertex set ${v_1,v_2,ldots,v_n}$, in which two vertices are adjacent if and only they have at least one common neighbor in the graph $G$. The basic properties of $con(G)$ and of its energy are established.

‎a set $s$ of vertices of a graph $g=(v,e)$ without isolated vertex‎ ‎is a {em total dominating set} if every vertex of $v(g)$ is‎ ‎adjacent to some vertex in $s$‎. ‎the {em total domatic number} of‎ ‎a graph $g$ is the maximum number of total dominating sets into‎ ‎which the vertex set of $g$ can be partitioned‎. ‎we show that the‎ ‎total domatic number of a random $r$-regular graph is almost‎...

a set $s$ of vertices in a graph $g=(v,e)$ is called a total$k$-distance dominating set if every vertex in $v$ is withindistance $k$ of a vertex in $s$. a graph $g$ is total $k$-distancedomination-critical if $gamma_{t}^{k} (g - x) < gamma_{t}^{k}(g)$ for any vertex $xin v(g)$. in this paper,we investigate some results on total $k$-distance domination-critical of graphs.

a vertex irregular total k-labeling of a graph g with vertex set v and edge set e is an assignment of positive integer labels {1, 2, ..., k} to both vertices and edges so that the weights calculated at vertices are distinct. the total vertex irregularity strength of g, denoted by tvs(g)is the minimum value of the largest label k over all such irregular assignment. in this paper, we study the to...

In 1963, Corrádi and Hajnal proved that for all k ≥ 1 and n ≥ 3k, every (simple) graph G on n vertices with minimum degree δ(G) ≥ 2k contains k disjoint cycles. The degree bound is sharp. Enomoto and Wang proved the following Ore-type refinement of the Corrádi-Hajnal Theorem: For all k ≥ 1 and n ≥ 3k, every graph G on n vertices contains k disjoint cycles, provided that d(x) + d(y) ≥ 4k − 1 for...

The second Zagreb coindex is a well-known graph invariant defined as the total degree product of all non-adjacent vertex pairs in a graph. The second Zagreb eccentricity coindex is defined analogously to the second Zagreb coindex by replacing the vertex degrees with the vertex eccentricities. In this paper, we present exact expressions or sharp lower bounds for the second Zagreb eccentricity co...

A graph G is equitably k-choosable if for any k-uniform list assignment L, G is L-colorable and each color appears on at most d|V (G)|/ke vertices. A graph G is equitable kcolorable if G has a proper vertex coloring with k colors such that the size of the color classes differ by at most 1. In this paper, we prove that if G is a planar graph without 5and 7-cycles, then G is equitably k-choosable...

In graph theory a partition of the vertex set of a graph is called equitable if for all pairs of cells all vertices in one cell have an equal number of neighbours in the other cell. Considering the implications for the adjacency matrix one may generalize that concept as a block partition of a complex square matrix s.t. each block has constant row sum. It is well known that replacing each block ...

A graph has an equitable, defective k-coloring (an ED-k-coloring) if there is a kcoloring of V (G) that is defective (every vertex shares the same color with at most one neighbor) and equitable (the sizes of all color classes differ by at most one). A graph may have an ED-kcoloring, but no ED-(k + 1)-coloring. In this paper, we prove that planar graphs with minimum degree at least 2 and girth a...

recently, hua et al. defined a new topological index based on degrees and inverse ofdistances between all pairs of vertices. they named this new graph invariant as reciprocaldegree distance as 1{ , } ( ) ( ( ) ( ))[ ( , )]rdd(g) = u v v g d u  d v d u v , where the d(u,v) denotesthe distance between vertices u and v. in this paper, we compute this topological index forgrassmann graphs.