The Randić index of an organic molecule whose molecular graph is G is the sum of the weights (d(u)d(v))− 1 2 of all edges uv of G, where d(u) and d(v) are the degrees of the vertices u and v in G. In the paper, we give a sharp lower bound on the Randić index of cacti.

The connectivity index wα(G) of a graph G is the sum of the weights (d(u)d(v)) of all edges uv of G, where α is a real number (α 6= 0), and d(u) denotes the degree of the vertex u. Let T be a tree with n vertices and k pendant vertices. In this paper, we give sharp lower and upper bounds for w1(T ). Also, for −1 ≤ α < 0, we give a sharp lower bound and a upper bound for wα(T ).

function f: G  , with this property that f(G1) = f(G2) if G1 and G2 are isomorphic. There are several vertex distance-based and degree-based indices which introduced to analyze the chemical properties of molecule graph. For instance: Wiener index, PI index, Szeged index, geometric-arithmetic index, atom-bond connectivity index and general sum connectivity index are introduced to test the perf...

 Mobile sensor networks rely heavily on inter-sensor connectivity for collection of data. Nodes in these networks monitor different regions of an area of interest and collectively present a global overview of some monitored activities or phenomena. A failure of a sensor leads to loss of connectivity and may cause partitioning of the network into disjoint segments. A number of approaches have be...

The Wiener index W (G) of a connected graph G is defined to be the sum

Let G be a graph and δv the degree of its vertex v . The connectivity index of G is χ = ∑ (δu δv) −1/2 , with the summation ranging over all pairs of adjacent vertices of G . We offer a simple proof that (a) among n-vertex graphs without isolated vertices, the star has minimal χvalue, and (b) among n-vertex graphs, the graphs in which all components are regular of non–zero degree have maximal (...

Polynomial interpolation can be used to obtain closed formulas for topological indices of infinite series of molecular graphs. The method is discussed and its advantages and limitations are pointed out. This is illustrated on fullerenes C12k+4 and four topological indices: the Wiener index, the edge Wiener index, the eccentric connectivity index, and the reverse Wiener index. The results for th...

The atom-bond connectivity (ABC) index, introduced by Estrada et al. in 1998, displays an excellent correlation with the formation heat of alkanes. We give upper bounds for this graph invariant using the number of vertices, the number of edges, the Randić connectivity indices, and the first Zagreb index. We determine the unique tree with the maximum ABC index among trees with given numbers of v...

Let G be a connected simple (molecular) graph. The distance d(u, v) between two vertices u and v of G is equal to the length of a shortest path that connects u and v. In this paper we compute some distance based topological indices of H-Phenylenic nanotorus. At first we obtain an exact formula for the Wiener index. As application we calculate the Schultz index and modified Schultz index of this...