On Generalized Edge Corona Product of Graphs

Let $G$ be a simple graph with $m$ edges and $H_i$, $1\le i\le m$ be simple graphs too. The generalized edge corona product of graphs $G$ and $H_1, …, H_m$,denotedby$G \diamond (H_1, …, H_m)$, is obtained by taking one copy of graphs $G$, $H_1,\mathrm{.}\mathrm{.}\mathrm{.},H_m$ and joining two end vertices of $i$-th edge of $G$ to every vertex of $H_i$, $1\le i\le m$. In this paper, some results regarding the $k$-distance chromatic number of Generalized edge corona product of graphs are presented. Also, as a consequence of our results, we compute this invariant for the graphs $K_n\diamond (H_1,\mathrm{.}\mathrm{.}\mathrm{.},H_m)$, $T\diamond (H_1, …, H_m)$and$K_{m,n} \diamond (H_1, …, H_m)$. Moreover, the domination set, domination number and the independence number of any connected graph $G$ and arbitrary graphs $H_i$, $1\le i\le \mathrm{\mid }E(G)\mathrm{\mid }$, are evaluated under generalized edge corona operation.