On Stress of a Vertex in a Graph

TL;DR

This paper studies the concept of vertex stress in graphs, characterizes stress regular graphs, and computes stress for various standard and special graph classes, revealing structural properties related to geodesics passing through vertices.

Contribution

It introduces the notion of stress regularity, characterizes graphs with specific stress properties, and computes stress for classes like diameter-2 graphs and corona products.

Findings

Strongly regular graphs are stress regular.

Characterization of 0, 1, 2-stress regular graphs.

Stress values computed for diameter-2 graphs and corona products.

Abstract

The stress of a vertex in a graph is the number of geodesics passing through it (A. Shimbel, 1953). A graph is $k$-stress regular if stress of each of its vertices is $k$. In this paper, we investigate some results and compute stress of vertices in some standard graphs and give a characterization of graphs with all vertices of zero stress except for one. Also we compute stress of vertices in graphs of diameter 2 and in the corona product $K_m \circ G$. Further we prove that any strongly regular graph is stress regular and characterize $k$-stress regular graphs for $k=0,1,2$.