On the $d_2$-Splitting Graph of a Graph


N. R. Santhi Maheswari, C. Sekar




For a positive integer $d$ and a vertex $v$ of a graph $G$, the $d^{th}$ degree of $v$ in $G$, denoted by $d_d (v)$, is defined as the number of vertices at a distance $d$ away from $v$. Hence $d_1(v) = d(v)$ and $d_2(v)$ means number of vertices at a distance 2 away from $v.$ A graph $G$ is said to be $(2,k )$-regular if $d_2 (v) = k$, for all $v$ in $G.$ In this paper we define $d_2$-splitting graph of a graph and we study some properties of $d_2$-splitting graph.