Geodesic
On the $d_2$-Splitting Graph of a Graph
Kragujevac Journal of Mathematics, Tome 36 (2012) no. 2, p. 315

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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.
Classification : 05C12
Keywords: Complete graph, trees, bipartite, complete bipartite, Eulerian graph, Hamiltonian graph, Connectivity of a graph, $(2,k)$ regular graph, $(d,k)$ regular graph 
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     author = {N. R. Santhi Maheswari and C. Sekar},
     title = {On the $d_2${-Splitting} {Graph} of a {Graph}},
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N. R. Santhi Maheswari; C. Sekar. On the $d_2$-Splitting Graph of a Graph. Kragujevac Journal of Mathematics, Tome 36 (2012) no. 2, p. 315 . http://geodesic.mathdoc.fr/item/KJM_2012_36_2_a14/