Study on total irregularity of graphs

Abstract

The total irregularity of a graph G is defined by For an ordered degree sequence of V(G) = {v1,v2, …, vn} with d(v1) ≤ d(v2) ≤ … ≤ d(vn), irrt(G) can be expressed in the form where dG(x) is the degree of x∈V(G). An edge e∈E(G) is said to be total irregular positive (negative, stable) inner edge if irrt>irrt(G) (irrt,irrt= irrt(G) respectively). Total irregular positive inner edge is denoted by TIPI edge. Similarly we use the notations TINI, TISI suitably. A graph G is called total irregular positive (negative, stable) inner graph if all the edges e∈E(G) are total irregular positive (negative, stable) inner edges; otherwise G is called a total irregular mixed inner graph.Total irregular positive inner graph is denoted by TIPI graph. Similarly we use the notations TINI, TISI suitably. In this paper, we prove that the Complete graph Kmn,m ≠ n and m,n ≥ 2 is a TIPI graph and the Star graph Sn n ≥ 3 is a TINI graph.