arXiv subjects
Publications and source records attributed to Sunilkumar M. Hosamani.
The study of graphs associated with of various algebraic structures is an emerging topic in algebraic graph theory. Recently, the concept of nonzero component graph of a finite dimensional vector space $Γ(\mathbb{V})$ was put forward by Das \cite{5}. In this paper, we study some degree based topological indices over $Γ(\mathbb{V})$ the derived graphs of $Γ(\mathbb{V})$.
The Mostar index of a connected graph \(G\) is defined as \[ Mo(G)=\sum_{uv\in E(G)}\bigl|n_u(uv)-n_v(uv)\bigr|, \] where for an edge \(e=uv\), \(n_u(e)\) denotes the number of vertices of \(G\) that are closer to \(u\) than to \(v\). In this paper, we determine the maximum possible Mostar index among all connected graphs of order \(n\) with exactly \(k\) cut edges, where \(1\le k\le n-1\). We prove that the maximum value is given by \(k(n-2)+(n-k-1)k\), and the unique extremal graph is \(K_{n-k}^k\) (a complete graph on \(n-k\) vertices with \(k\) pendant edges attached to a single vertex). We also establish a sharp lower bound and characterise the extremal graphs for the minimum value. Furthermore, we extend the results to graphs with a given cyclomatic number and a given number of cut edges. Our findings complete the extremal characterisation of the Mostar index for this fundamental graph class.
The bond incident degree (BID) index of a graph \(G\) is defined as \(\BID(G) = \sum_{u_1u_2\in E(G)} f(d(u_1), d(u_2))\), where \(f(x,y)=f(y,x)\) is a real-valued function. In this paper, using graph transformation methods, we establish the maximum bond incident degree indices of trees and unicyclic graphs with a fixed diameter for the inverse sum indeg (ISI) index. The ISI index corresponds to the function \(f(x,y) = \frac{xy}{x+y}\). We prove that for trees \(T \in \mathbb{T}_{n,d}\) with \(d \geq 3\) and \(n \geq d+3\), the maximum ISI index is attained by the tree \(T_{n,d}^*\). For unicyclic graphs, we characterize the extremal graphs for diameters \(d=2\), \(d=3\), and \(d \geq 4\). Specifically, the maximum ISI index is achieved by \(S_n^+\) for \(d=2\), by \(C_n^*\) for \(d=3\), and by \(\mathcal{U}_{n,d}\) for \(d \geq 4\).
Inspired by recent work on anti-covid-19 drugs \cite{2} here we study the Quantitative-structure property relationships(QSPR) of phytochemicals screened against SARS-CoV-2 $3CL^{pro}$ with the help of topological indices like the first Zagreb index $M_{1}$, second Zagreb index $M_{2}$, Randi$\acute{c}$ index $R$, Balban index $J$ and sum-connectivity index $SCI(G)$. Our study has raveled that the sum-connectivity index $(SCI)$ and the first Zagreb index $(M_{1})$ are two important parameters to predict the molecular weight and the topological polar surface area of phytochemicals respectively.