On $b$-Eccentricity in Graphs

Abstract

The distance $d(u,v)$ between a pair of vertices $u$ and $v$ in a connected graph $G$ is the length of a shortest path joining them. A vertex $v$ of a connected graph $G$ is an eccentric vertex of a vertex $u$ if $v$ is a vertex at greatest distance from $u$; while $v$ is an eccentric vertex of $G$ if $v$ is an eccentric vertex of some vertex of $G$. A vertex $v$ of $G$ is a boundary vertex of a vertex $u$ if $d(u,w)\le d(u,v)$ for each neighbour $w$ of $v$. A vertex $v$ is a boundary vertex of $G$ if $v$ is a boundary vertex of some vertex of $G$. It is easy to see that for a vertex $u$, its eccentric vertices are boundary vertices for $u$; but not conversely. In this paper, we introduce a new type of eccentricity called b-eccentricity and we study its properties.