In this paper, we present some new lower and upper bounds for the modified Randic index in terms of maximum, minimum degree, girth, algebraic connectivity, diameter and average distance. Also we obtained relations between this index with Harmonic and Atom-bond connectivity indices. Finally, as an application we computed this index for some classes of nano-structures and linear chains.

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...

A two-mode network is a type of in which nodes can be divided into two sets such way that links established between different types nodes. The relationship separate entities modeled as bipartite network. In computer networks data transmitted form packets source to destination. Such packet-switched rely on routing protocols select the best path. Configurations these depends acquirements; why one...

The Randić index of a graph G, written R(G), is the sum of 1 √ d(u)d(v) over all edges uv in E(G). Let d and D be positive integers d < D. In this paper, we prove that if G is a graph with minimum degree d and maximum degree D, then R(G) ≥ √ dD d+Dn; equality holds only when G is an n-vertex (d,D)-biregular. Furthermore, we show that if G is an n-vertex connected graph with minimum degree d and...

The Randić index R(G) of a graph G is defined by R(G) = ∑ uv 1 √ d(u)d(v) , where d(u) is the degree of a vertex u in G and the summation extends over all edges uv of G. Aouchiche et al. proposed a conjecture on the relationship between the Randić index and the diameter: for any connected graph on n ≥ 3 vertices with the Randić index R(G) and the diameter D(G), R(G) − D(G) ≥ √ 2 − n+1 2 and R(G...

let $d$ be a digraph with vertex set $v(d)$. for vertex $vin v(d)$, the degree of $v$,denoted by $d(v)$, is defined as the minimum value if its out-degree and its in-degree.now let $d$ be a digraph with minimum degree $deltage 1$ and edge-connectivity$lambda$. if $alpha$ is real number, then the zeroth-order general randic index is definedby $sum_{xin v(d)}(d(x))^{alpha}$. a digraph is maximall...

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: