Partial differentiability on graphs

Partial differentiability on graphs

**Authors : **

M. Regees^{ 1} and T. Sajitha Kumari ^{2 *}

**Author Address : **

^{1} Department of Mathematics, Malankara Catholic College, Kaliakavilai-629153, Tamil Nadu, India.

^{2} Research Scholar, Department of Mathematics, Scott Christian College, Nagercoil- 629001, Tamil Nadu, India.

*Corresponding author.

**Abstract : **

Operations play a vital role in the field of Mathematics. There are many operations by which new graphs are obtained from the old ones. In this paper we try to form a new graph from the old one through a new operation, partial differentiability. Let $G$ be a graph of order $n$. Consider an arbitrary vertex $v$ in $G$. We remove all edges which are adjacent to the vertex $v$. The resultant graph is denoted by $G_{v}^{(1)}$. This is called the partial differentiation of $G$ with respect to the vertex $v$. Now we consider another vertex $u$ in $G_{v}^{(1)}$ and remove all edges which are adjacent to $u$. The resultant graph is denoted by $G_{u}^{(2)}$. This is the partial derivative of $G_{v}^{(1)}$ with respect to $u$. The minimum of $r$ such that $G^{(r)}cong mK_1$, is called the order of partial differentiation, denoted by $r(G)$, where $m$ is a positive integer. In this paper we introduced the partial differentiability of graphs.

**Keywords : **

Graph, Differentiation, Partial differentiation.

**DOI : **

**Article Info : **

*Received : * December 21, 2018; *Accepted : * February 11, 2019.