Superior Distance in Graphs

Abstract

The main objective of this paper is to introduce a generalization of distance called superior distance in graphs. For two vertices $u$ and $v$ of a connected graph, we define ${\text{D}}_{u,v}=\text{N}[u]\cup \text{N}[v]$. We define a ${\text{D}}_{u,v}$-walk as a $u$-$v$ walk that contains every vertex of ${\text{D}}_{u,v}$. The superior distance ${\text{d}}_D(u,v)$ from $u$ to $v$ is the length of a shortest ${\text{D}}_{u,v}$-walk. In this paper, first we give the bounds for the superior diameter of a graph and a property that relates the superior eccentricities of adjacent vertices. Finally, we investigate those graphs that are isomorphic to the superior center of some connected graph and those graphs that are isomorphic to the superior periphery of some connected graph.