The Padmakar-Ivan (PI) index is a first-generation topological index (TI) based on sums over all edges between numbers of edges closer to one endpoint and numbers of edges closer to the other endpoint. Edges at equal distances from the two endpoints are ignored. An analogous definition is valid for the Wiener index W, with the difference that sums are replaced by products. A few other TIs are d...

In this paper, we present some inequalities for the Co-PI index involving the some topological indices, the number of vertices and edges, and the maximum degree. After that, we give a result for trees. In addition, we give some inequalities for the largest eigenvalue of the Co-PI matrix of G.

let $a$ be a non-trivial abelian group and $a^{*}=asetminus {0}$. a graph $g$ is said to be $a$-magic graph if there exists a labeling$l:e(g)rightarrow a^{*}$ such that the induced vertex labeling$l^{+}:v(g)rightarrow a$, define by $$l^+(v)=sum_{uvin e(g)} l(uv)$$ is a constant map.the set of all constant integerssuch that $sum_{uin n(v)} l(uv)=c$, for each $vin n(v)$,where $n(v)$ denotes the s...

The hB-pi tree is a multi-attribute index method which combines the hB tree, a multi-attribute access structure and the II-tree, an abstract index with efficient concurrency and recovery methods. The hB-pi tree is a balanced tree and adapts well to different data distributions. Furthermore, the hB-pi tree inherits the efficient concurrency and recovery methods from the II tree. This paper provi...

In the present work, we investigate decays of $B^{0}_{s} \rightarrow \phi\pi^{+}\pi^{-}$ and $B^{0} with final state interactions based on chiral unitary approach. $\pi^+\pi^-$ its coupled channels, study effects $\eta\eta$ channel in two-body for reproduction $f_{0}(980)$ state. Our results invariant mass distributions decay describe experimental data up to 1 GeV well, resonance contributions ...

The Padmakar-Ivan (PI) index is a Wiener-Szeged-like topological index, which reflects certain structural features of organic molecules. In this paper, we study the maximum PI indices and the minimum PI indices for trees and unicyclic graphs respectively.

the eccentricity connectivity index of a molecular graph g is defined as (g) = av(g)deg(a)ε(a), where ε(a) is defined as the length of a maximal path connecting a to othervertices of g and deg(a) is degree of vertex a. here, we compute this topological index forsome infinite classes of dendrimer graphs.

The atom-bond connectivity index is a recently introduced topological index defined as [Formula: see text], where du denotes degree of vertex u. Here we define a new version of the ABC index as [Formula: see text], where nu denotes the number of vertices of G whose distances to vertex u are smaller than those to other vertex v of the edge e = uv, and nv is defined analogously. The goal of this ...

The PI index is a graph invariant defined as the summation of the sums of edges of neu and nev over all the edges of connected graph G, where neu is the number of edges of G lying closer to u than to v and nev is the number of edges of G lying closer to v than to u. The index is very simple to calculate and has disseminating power similar to that of the Wiener and the Szeged indices. The compre...

The atom bond connectivity index of a graph is a new topological index was defined by E. Estrada as ABC(G)  uvE (dG(u) dG(v) 2) / dG(u)dG(v) , where G d ( u ) denotes degree of vertex u. In this paper we present some bounds of this new topological index.