Mathematical and chemistry properties of geometry-based invariants

Recently, based on elementary geometry, Gutman proposed several geometry-based invariants (i.e., $SO$, $SO_{1}$, $SO_{2}$, $SO_{3}$, $SO_{4}$, $SO_{5}$, $SO_{6}$). The Sombor index was defined as $SO(G)=\sum\limits_{uv\in E(G)}\sqrt{d_{u}^{2}+d_{v}^{2}}$, the first Sombor index was defined as $SO_{1}(G)= \frac{1}{2}\sum\limits_{uv\in E(G)}|d_{u}^{2}-d_{v}^{2}|$, where $d_{u}$ denotes the degree of vertex $u$. In this paper, we consider the mathematical and chemistry properties of these geometry-based invariants. We determine the maximum trees (resp. unicyclic graphs) with given diameter, the maximum trees with given matching number, the maximum trees with given pendent vertices, the maximum trees (resp. minimum trees) with given branching number, the minimum trees with given maximum degree and second maximum degree, the minimum unicyclic graphs with given maximum degree and girth, the minimum connected graphs with given maximum degree and pendent vertices, and some properties of maximum connected graphs with given pendent vertices with respect to the first Sombor index $SO_{1}$. As an application, we inaugurate these geometry-based invariants and verify their chemical applicability. We used these geometry-based invariants to model the acentric factor (resp. entropy, enthalpy of vaporization, etc.) of alkanes, and obtained satisfactory predictive potential, which indicates that these geometry-based invariants can be successfully used to model the thermodynamic properties of compounds.Comment: 29 pages, 11 figur

Eccentric connectivity index

The eccentric connectivity index $\xi^c$ is a novel distance--based molecular structure descriptor that was recently used for mathematical modeling of biological activities of diverse nature. It is defined as $\xi^c (G) = \sum_{v \in V (G)} deg (v) \cdot \epsilon (v)$\,, where $deg (v)$ and $\epsilon (v)$ denote the vertex degree and eccentricity of $v$\,, respectively. We survey some mathematical properties of this index and furthermore support the use of eccentric connectivity index as topological structure descriptor. We present the extremal trees and unicyclic graphs with maximum and minimum eccentric connectivity index subject to the certain graph constraints. Sharp lower and asymptotic upper bound for all graphs are given and various connections with other important graph invariants are established. In addition, we present explicit formulae for the values of eccentric connectivity index for several families of composite graphs and designed a linear algorithm for calculating the eccentric connectivity index of trees. Some open problems and related indices for further study are also listed.Comment: 25 pages, 5 figure

The quotients between the (revised) Szeged index and Wiener index of graphs

Let $Sz(G),Sz^*(G)$ and $W(G)$ be the Szeged index, revised Szeged index and Wiener index of a graph $G.$ In this paper, the graphs with the fourth, fifth, sixth and seventh largest Wiener indices among all unicyclic graphs of order $n\geqslant 10$ are characterized; as well the graphs with the first, second, third, and fourth largest Wiener indices among all bicyclic graphs are identified. Based on these results, further relation on the quotients between the (revised) Szeged index and the Wiener index are studied. Sharp lower bound on $Sz(G)/W(G)$ is determined for all connected graphs each of which contains at least one non-complete block. As well the connected graph with the second smallest value on $Sz^*(G)/W(G)$ is identified for $G$ containing at least one cycle.Comment: 25 pages, 5 figure

On the third ABC index of trees and unicyclic graphs

Let $G=(V,E)$ be a simple connected graph with vertex set $V(G)$ and edge set $E(G)$. The third atom-bond connectivity index, $ABC_3$ index, of $G$ is defined as $ABC_3(G)=\sum\limits_{uv\in E(G)}\sqrt{\frac{e(u)+e(v)-2}{e(u)e(v)}}$, where eccentricity $e(u)$ is the largest distance between $u$ and any other vertex of $G$, namely $e(u)=\max\{d(u,v)|v\in V(G)\}$. This work determines the maximal $ABC_3$ index of unicyclic graphs with any given girth and trees with any given diameter, and characterizes the corresponding graphs

On the positive and negative inertia of weighted graphs

The number of the positive, negative and zero eigenvalues in the spectrum of the (edge)-weighted graph $G$ are called positive inertia index, negative inertia index and nullity of the weighted graph $G$, and denoted by $i_+(G)$, $i_-(G)$, $i_0(G)$, respectively. In this paper, the positive and negative inertia index of weighted trees, weighted unicyclic graphs and weighted bicyclic graphs are discussed, the methods of calculating them are obtained.Comment: 12. arXiv admin note: text overlap with arXiv:1107.0400 by other author