### 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 :

10.26637/MJM0S01/0025

#### Article Info :

Received : December 21, 2018; Accepted : February 11, 2019. Search Information Pages

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