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 $(d d s)$ of a vertex $v$ in a graph $G$ is a list of the number of vertices at distance $1,2, \ldots, e(v)$; in that order, where $e(v)$ denotes the eccentricity of $v$ in $G$. Thus, the sequence $\left(d_{i_0}, d_{i_1}, d_{i_2}, \ldots, d_{i_j}, \ldots\right)$ is 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.