arXiv subjects

A. Bharali

Publications and source records attributed to A. Bharali.

3 recordsLinked to original sources

arXiv2025

On Randi\'c energy of a vertex

In 2018, Arizmendi and Juarez introduced the concept of energy of a vertex, a novel approach allowing the total energy of a graph to be expressed as the sum of the energies of its individual vertices. In this article, we extend the notion of energy of a vertex to the context of the Randi\'c matrix. We define the Randi\'c energy of a vertex and explore its mathematical properties through various combinatorial techniques. We derive several upper and lower bounds for the Randi\'c energy of a vertex. Furthermore, we establish that among the connected graphs, the central vertex of a star attains the maximum Randi\'c energy, whereas pendent vertices attain the minimum. Also, we report the Coulson-type integral formula for the Randi\'c energy of a vertex and its applications.

math.SP↗

arXiv2023

Predictive ability comparison of different versions of some well known degree dependent topological indices

To avoid extensive lab work on properties of chemical compounds, QSPR/QSAR analysis for topological descriptors is a productive statistical approach to analyze various physicochemical properties of chemical compounds. Many researchers have investigated on correlation of degree-based topological descriptors. In this article we present the predictive ability of 6 well known degree dependant topological indices of 22 lower poly cyclic aromatic hydrocarbons in three versions, and we have done a comparison analysis of three versions of considered topological indices for their predictive ability.

physics.chem-ph↗

arXiv2023

On the eigenvalues and energy of the $A_α$-matrix of graphs

For a graph $G$, the generalized adjacency matrix $A_α(G)$ is the convex combination of the diagonal matrix $D(G)$ and the adjacency matrix $A(G)$ and is defined as $A_α(G)=αD(G)+(1-α) A(G)$ for $0\leq α\leq 1$. This matrix has been found to be useful in merging the spectral theories of $A(G)$ and the signless Laplacian matrix $Q(G)$ of the graph $G$. The generalized adjacency energy or $A_α$-energy is the mean deviation of the $A_α$-eigenvalues of $G$ and is defined as $E(A_α(G))=\sum_{i=1}^{n}|p_i-\frac{2αm}{n}|$, where $p_i$'s are $A_α$-eigenvalues of $G$. In this paper, we investigate the $A_α$-eigenvalues of a strongly regular graph $G$. We observe that $A_α$-spectral radius $p_1$ satisfies $δ(G)\leq p_1 \leq Δ(G)$, where $δ(G)$ and $Δ(G)$ are, respectively, the smallest and the largest degrees of $G$. Further, we show that the complete graph is the only graph to have exactly two distinct $A_α$-eigenvalues. We obtain lower and upper bounds of $A_α$-energy in terms of order, size and extremal degrees of $G$. We also discuss the extremal cases of these bounds.

math.SP↗