New results on distance degree sequences of graphs

Abstract

The distance d(u, v) from a vertex u of G to a vertex v is the length of a shortest u to v path. The distance degree sequence (dds) of a vertex v in a graph G is a list of the number of vertices at distance 1, 2, . . . . , e(v); in that order, where e(v) denotes the eccentricity of v in G. Thus, the sequenceis the distance degree sequence of the vertex v i in G where, d i j denotes the number of vertices at distance j from v i . In this article we present results to find distance degree sequences of some of the derived graphs viz., the line graph, the sub-division graph, the total graph, the powers of a graph, the Mycieleskian of a graph etc.

Computation of Mostar index for some graph operations

“…The two distance‐based sequences, the distance degree sequence and path degree sequences, were studied by Randić [51] for chemical graph theory. A comprehensive study has been accomplished in this area of research by Bloom et al [52, 53], Huilgol et al [54–60], Slater [61], Quintas et al [62], and so on. Now, we discuss some known results that will be used in the rest of the paper.…”