Proofs of conjectures on the Randi? index and average eccentricity ?
The Randi? index R(G) R ( G ) of a graph G G is defined by R ( G ) = ? u v 1 d ( u ) d ( v ) , where d(u) d ( u ) is the degree of a vertex u u and the summation extends over all edges uv u v of G G . The eccentricity ?(v) ? ( v ) of a vertex v v is the maximum distance from it to any other vertex and the average eccentricity ? ? ( G ) of graph G G is the mean value of eccentricities of all vertices of G G . There are relations between the Randi? index and the average eccentricity of connected graphs conjectured by a computer program called AGX: for any connected graph G G on n?14 n ? 14 vertices, both lower bounds of R ( G ) + ? ? ( G ) and R ( G ) ? ? ? ( G ) are achieved only by a star. In this paper, we show that both conjectures are true.
conjecture randi? index average eccentricity