Reciprocal distance energy of complete multipartite graphs

TL;DR

This paper calculates the reciprocal distance energy of complete multipartite graphs, compares energies among different graph types, and studies how edge deletion affects this energy.

Contribution

It introduces formulas for reciprocal distance energy of complete multipartite graphs and analyzes energy variations under edge deletion.

Findings

Complete split graph has minimum reciprocal distance energy among k-partite graphs.

Turan graph has maximum reciprocal distance energy among k-partite graphs.

Edge deletion decreases reciprocal distance energy in certain bipartite graphs.

Abstract

In this paper, first we compute the energy of a special partitioned matrix under some cases. As a consequence, we obtain the reciprocal distance energy of the complete multipartite graph and also we give various other energies of complete multipartite graphs. Next, we show that among all complete k-partite graphs on n vertices, the complete split graph has minimum reciprocal distance energy and the reciprocal distance energy is maximum for the Turan graph. At last, it is shown that the reciprocal distance energy of the complete bipartite graph km,m decreases under deletion of an edge if 2\le m\le 7, whereas the reciprocal distance energy increases if 8 \le m. Also, we show that the reciprocal distance energy of the complete tripartite graph does not increase under edge deletion