On a conjecture of the Randi? index
The Randi? index of a graph G G is defined as R ( G ) = ? u ? v ( d ( u ) d ( v ) ) ? 1 2 , where d(u) d ( u ) is the degree of vertex u u and the summation goes over all pairs of adjacent vertices u u , v v . A conjecture on R(G) R ( G ) for connected graph G G is as follows: R(G)?r(G)?1 R ( G ) ? r ( G ) ? 1 , where r(G) r ( G ) denotes the radius of G G . We proved that the conjecture is true for biregular graphs, connected graphs with order n?10 n ? 10 and tricyclic graphs.
conjecture randi? index connected graphs with order ? 10