Proofs of the AutoGraphiX conjectures on the domination number, average eccentricity and proximity

Abstract

The eccentricity of a vertex is the greatest distance from it to any other vertex and the average eccentricity of a graph G is the average value of eccentricities of all vertices of G. The proximity of a vertex in a connected graph is the average distance from it to all other vertices and the proximity of a connected graph G is the minimum average distance from a vertex of G to all others. A set S⊆V(G) is called a dominating set of G if NG(x)⋂S≠0̸ for any vertex x∈V(G)∖S. The domination number γ(G) of G is the minimum cardinality of all dominating sets of G. In this paper, we improve and prove two AutoGraphiX conjectures. One gives the sharp upper bound on the quotient of the domination number and average eccentricity, and another shows the sharp upper bound about the difference between the domination number and proximity.