The diameter of a connected graph $G$, denoted by $diam(G)$, is the maximum distance between any pair of vertices of $G$. Let $L(G)$ be the line graph of $G$. We establish necessary and sufficient conditions under which for a given integer $k geq 2$, $diam(L(G)) leq k$.

For a nontrivial connected graph G with no isolated vertex, nonempty subset D \(\subseteq\) V (G) is rings dominating set if and for each vertex \(\upsilon\) \(\in\) \ adjacent to at least two vertices in D. Thus, the of all D, \(\mid\)N(\(\upsilon\)) \(\cap\) (V D)\(\mid\) \(\ge\) 2. Moreover, called minimum smallest size given graph. The cardinality domination number G, denoted by \(\gamma\)r...

An r-dynamic coloring of a graph G is proper such that every vertex in V(G) has neighbors at least $\min\{d(v),r\}$ different color classes. The chromatic number denoted as $\chi_r (G)$, the k coloring. In this paper we obtain central graph, middle total line para-line and sub-division comb $P_n\odot K_1$ by $C(P_n\odot K_1), M(P_n\odot T(P_n\odot L(P_n\odot P(P_n\odot K_1)$ $S(P_n\odot respect...

‎let $g=(v(g),e(g))$ be a simple connected graph with vertex set $v(g)$ and edge‎ ‎set $e(g)$‎. ‎the (first) edge-hyper wiener index of the graph $g$ is defined as‎: ‎$$ww_{e}(g)=sum_{{f,g}subseteq e(g)}(d_{e}(f,g|g)+d_{e}^{2}(f,g|g))=frac{1}{2}sum_{fin e(g)}(d_{e}(f|g)+d^{2}_{e}(f|g)),$$‎ ‎where $d_{e}(f,g|g)$ denotes the distance between the edges $f=xy$ and $g=uv$ in $e(g)$ and $d_{e}(f|g)=s...