The Eccentric-Distance Sum of Cycles and Related Graphs

Abstract

Let G = (V, E) be a simple connected graph. The eccentric-distance sum of G is defined as ξ ds (G) = u∈V (G) e(u)D(u) where e(u) is the eccentricity of the vertex u in G and D(u) is the sum of distances between u and all other vertices of G. In this paper, we establish formulae to calculate the eccentric-distance sum for some cycle related graphs, namely Cn, complement of Cn, shadow of Cn and the line graph of Cn. Also, it is shown that, the eccentric-distance sum of Cn is less than the eccentric-distance sum of shadow of Cn for all n ≥ 3.