The general Randić index Rα(G) of a graph G is defined as the sum of the weights (d(u)d(v)) α of all edges uv of G, where d(u) denotes the degree of a vertex u in G and α is an arbitrary real number. Clark and Moon gave the lower and upper bounds for the Randić index R −1 of all trees, and posed the problem to determine better bounds. In this paper we give the best possible lower and upper boun...

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. We prove that for any tree T with n1 leaves R(T ) ≥ ad(T ) + max(0,n1 − 2), where ad(T ) is the average distance between vertices of T . As a consequence we resolve the conjecture R(G) ≥ ad(G) given by Fajtlowicz in 1988 for the case when G is a tree.

Let $G$ be a non-abelian group. The non-commuting graph $Gamma_G$ of $G$ is defined as the graph whose vertex set is the non-central elements of $G$ and two vertices are joined if and only if they do not commute.In this paper we study some properties of $Gamma_G$ and introduce $n$-regular $AC$-groups. Also we then obtain a formula for Szeged index of $Gamma_G$ in terms of $n$, $|Z(G)|$ and $|G|...

*Correspondence: [email protected] 1School of Natural Sciences (SNS), National University of Sciences and Technology (NUST), Sector H-12, Islamabad, Pakistan 2Department of Mathematical Sciences, United Arab Emirates University, P.O. Box 15551, Al Ain, United Arab Emirates Full list of author information is available at the end of the article Abstract Let G be a connected graph. The degree of...

The connectivity index χ1(G) of a graph G is the sum of the weights d(u)d(v) of all edges uv of G, where d(u) denotes the degree of the vertex u. Let T (n, r) be the set of trees on n vertices with diameter r . In this paper, we determine all trees in T (n, r) with the largest and the second largest connectivity index. Also, the trees in T (n, r) with the largest and the second largest connecti...

The eccentric connectivity index ξ is a distance–based molecular structure descriptor that was recently used for mathematical modeling of biological activities of diverse nature. We prove that the broom has maximum ξ among trees with a fixed maximum vertex degree, and characterize such trees with minimum ξ . In addition, we propose a simple linear algorithm for calculating ξ of trees.

If G is a connected graph with vertex set V (G), then the eccentric connectivity index of G, denoted by ξc(G), is defined as ∑ v∈V (G) deg(v)ec(v), where deg(v) is the degree of a vertex v and ec(v) is its eccentricity. Morgan et al. [5] investigated the eccentric connectivity index of trees. In this paper, we investigate the eccentric connectivity index of unicyclic graphs. Upper bound is obta...

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...

Connectivity is a vital element in landscape structure because of its importance in species–landscape interactions. Connectivity analysis of green spaces in urban landscapes, especially in high-density cities such as Hong Kong, differs from that of habitats in natural or rural landscapes. Using the human being as the target species, we formulated with GIS techniques a resistance weight, a struc...

We propose a new index, the Container Port Connectivity Index (CPCI), to measure the trade connectivity of ports within the network of container shipping. This index is based on both economics and network topology, and a distinctive feature is that the strength of a port is based on its position within the global structure of the shipping network and not just on local information such as the nu...