Distance strings of the vertices of certain graphs

Abstract

The notion of the distance string of a vertex \(v_i \in V(G)\) which is denoted by, \(\tau(v_i)\) is introduced. Distance strings permit a new approach to determining the induced vertex stress, the total induced vertex stress and total vertex stress (sum of vertex stress over all vertices) of a graph. A seemingly under-studied topic i.e. the eccentricity of a vertex of a bipartite Kneser graph \(BK(n,k)\), \(n \geq 2k + 1\) has been furthered. A surprisingly simple result was established, namely for \(k \geq 2\), \(diam(BK(n,k)) = 5\) if \(n = 2k + 1\) and \(diam(BK(n,k)) = 3\) if \(n \neq 2k + 1\).