On the Inequality between Radius and Randić Index for Graphs
نویسندگان
چکیده
The Randić index R(G) of a graph G is the sum of weights (deg(u) deg(v))−0.5 over all edges uv of G, where deg(v) denotes the degree of a vertex v. Let r(G) be the radius of G. We prove that for any connected graph G of maximum degree four which is not a path with even number of vertices, R(G) ≥ r(G). As a consequence, we resolve the conjecture R(G) ≥ r(G)− 1 given by Fajtlowicz in 1988 for the case when G is a chemical graph.
منابع مشابه
The Randić index and signless Laplacian spectral radius of graphs
Given a connected graph G, the Randić index R(G) is the sum of 1 √ d(u)d(v) over all edges {u, v} of G, where d(u) and d(v) are the degree of vertices u and v respectively. Let q(G) be the largest eigenvalue of the singless Laplacian matrix of G and n = |V (G)|. Hansen and Lucas (2010) made the following conjecture:
متن کاملOn the Modified Randić Index of Trees, Unicyclic Graphs and Bicyclic Graphs
The modified Randić index of a graph G is a graph invariant closely related to the classical Randić index, defined as
متن کاملSharp Bounds on the PI Spectral Radius
In this paper some upper and lower bounds for the greatest eigenvalues of the PI and vertex PI matrices of a graph G are obtained. Those graphs for which these bounds are best possible are characterized.
متن کاملOn the Randić index and Diameter of Chemical Graphs
Using the AutoGraphiX 2 system, Aouchiche, Hansen and Zheng [2] proposed a conjecture that the difference and the ratio of the Randić index and the diameter of a graph are minimum for paths. We prove this conjecture for chemical graphs.
متن کاملSome topological indices of graphs and some inequalities
Let G be a graph. In this paper, we study the eccentric connectivity index, the new version of the second Zagreb index and the forth geometric–arithmetic index.. The basic properties of these novel graph descriptors and some inequalities for them are established.
متن کامل