The Connectivity and the Harary Index of a Graph

The Harary index of a graph is defined as the sum of reciprocals of distances between all pairs of vertices of the graph. In this paper we provide an upper bound of the Harary index in terms of the vertex or edge connectivity of a graph. We characterize the unique graph with maximum Harary index among all graphs with given number of cut vertices or vertex connectivity or edge connectivity. In addition we also characterize the extremal graphs with the second maximum Harary index among the graphs with given vertex connectivity

On graphs with maximum Harary spectral radius

Let $G$ be a simple graph with vertex set $V(G) = \{v_1 ,v_2 ,\cdots ,v_n\}$. The Harary matrix $RD(G)$ of $G$, which is initially called the reciprocal distance matrix, is an $n \times n$ matrix whose $(i,j)$-entry is equal to $\frac{1}{d_{ij}}$ if $i\not=j$ and $0$ otherwise, where $d_{ij}$ is the distance of $v_i$ and $v_j$ in $G$. In this paper, we characterize graphs with maximum spectral radius of Harary matrix in three classes of simple connected graphs with $n$ vertices: graphs with fixed matching number, bipartite graphs with fixed matching number, and graphs with given number of cut edges, respectively.Comment: 12 page

Topological indices for the antiregular graphs

We determine some classical distance-based and degree-based topo- logical indices of the connected antiregular graphs (maximally irregular graphs). More precisely, we obtain explicitly the k-Wiener index, the hyper-Wiener index, the degree distance, the Gutman index, the first, sec- ond and third Zagreb index, the reduced first and second Zagreb index, the forgotten Zagreb index, the hyper-Zagreb index, the refined Zagreb index, the Bell index, the min-deg index, the max-deg index, the symmet- ric division index, the harmonic index, the inverse sum indeg index, the M-polynomial and the Zagreb polynomial

Mostar index of bridge graphs

Topological indices are the numerical descriptors of a molecular structure obtained via molecular graph G. Topological indices are used in the structure-property relationship, structure-activity relations, and nanotechnology. Also, they hold us to predict certain physicochemical properties such as boiling point, enthalpy of vaporization, stability, and so on. In this study, it is considered the Mostar index. It is present upper bound for Mostar index of bridge graphs. Moreover, it is given exact expressions for the Mostar index of bridge graphs of the path, star, cycle, and complete graphs.Publisher's Versio

On Topological Indices And Domination Numbers Of Graphs

Topological indices and dominating problems are popular topics in Graph Theory. There are various topological indices such as degree-based topological indices, distance-based topological indices and counting related topological indices et al. These topological indices correlate certain physicochemical properties such as boiling point, stability of chemical compounds. The concepts of domination number and independent domination number, introduced from the mid-1860s, are very fundamental in Graph Theory. In this dissertation, we provide new theoretical results on these two topics. We study k-trees and cactus graphs with the sharp upper and lower bounds of the degree-based topological indices(Multiplicative Zagreb indices). The extremal cacti with a distance-based topological index (PI index) are explored. Furthermore, we provide the extremal graphs with these corresponding topological indices. We establish and verify a proposed conjecture for the relationship between the domination number and independent domination number. The corresponding counterexamples and the graphs achieving the extremal bounds are given as well