arXiv subjects

Kinkar Chandra Das

Publications and source records attributed to Kinkar Chandra Das.

At least 19 recordsLinked to original sources

arXiv2026

Hoffman-type Results for the Sum of k Largest Eigenvalues of a Graph

Let $S_k(G)$ denote the sum of the $k$ largest eigenvalues of a graph $G$. Motivated by the classical Hoffman program for the spectral radius of a graph, we investigate an additive Hoffman-type problem for $S_k(G)$. For each fixed $k\geq 2$ and sufficiently large order $n$, we characterize all connected graphs satisfying $S_k(G)<2k$. As a consequence, we prove that the path $P_n$ is the unique minimizer of $S_k(G)$ among all connected graphs of order $n$. \vspace*{2mm} We further investigate the first Hoffman-type range \[ 2k\leq S_k(G)<2k+\sqrt{2+\sqrt5}-2. \] We completely characterize the non-tree graphs in this range and reduce the tree case to several explicit families. The proofs combine Ky Fan's variational principle, spectral estimates from vertex-disjoint subgraphs, structural results for graphs with small spectral radius, and long-path arguments for bounded-degree graphs.

math.CO↗

arXiv2026

Structural Properties and Applications of the Augmented Sombor Index

Topological indices are key quantitative descriptors in mathematical chemistry, unchanged under symmetry operations and retaining graph connectivity; they capture molecular structural features to provide insights into molecular stability and chemical properties, becoming indispensable in cheminformatics and theoretical chemistry. Among degree-based indices, the \textbf{Sombor index} is widely concerned for capturing structural information, and motivated by enhanced structural discrimination, the \textbf{augmented Sombor index} ($ASO$) is defined for a connected graph $Ω$ with $|V(Ω)|\geq 3$ as $$ASO(Ω) = \sum_{v_iv_j\in E(Ω)} \sqrt{\frac{d_i^2 + d_j^2}{d_i + d_j - 2}},$$ where $d_i$ and $d_j$ are the degrees of vertices $v_i$ and $v_j$, respectively. Within the scope of this study, we first establish several sharp bounds for the augmented Sombor index and characterize the extremal graphs attaining these bounds. In particular, we determine the minimum value of the $ASO$ index for unicyclic graphs with a prescribed girth and characterize all graphs achieving this minimum. We also identify the second maximum $ASO$ value among trees and characterize the corresponding extremal tree structures. Furthermore, the minimum and maximum values of the $ASO$ index for bipartite graphs and chemical graphs are obtained, together with a complete characterization of the associated extremal graphs. In addition, we characterize the chemical trees that maximize the $ASO$ index. The chemical applicability of the $ASO$ index is investigated through quantitative structure-property relationship (QSPR) analysis, supported by a comparative assessment of several variants of the Sombor index. Finally, we present concluding remarks and outline potential directions for future research on the augmented Sombor index of graphs.

math.CO↗

arXiv2026

Sum of the $k$ Largest Eigenvalues of Symmetric Matrices: Theory and Applications

This paper establishes new upper bounds for the sum of the $k$ largest eigenvalues of symmetric matrices. When applied to the adjacency matrix of a graph, our results improve upon a related bound due to Mohar {\bf [On the sum of k largest eigenvalues of graphs and symmetric matrices, J. Combin. Theory Ser. B 99 (2009) 306--313]}. Furthermore, in the case of the Laplacian matrix, we prove that the well-known Brouwer's conjecture {\bf [Spectra of Graphs, Springer, New York, 2012]} holds for small values of $k$ for almost all graphs, thereby taking a significant step toward its complete resolution.

math.CO↗

arXiv2026

Resolving Open Problems on the Hyper-Zagreb Index and its Chemical Applications

Topological indices are numerical invariants derived from molecular graphs and play an important role in characterizing chemical compounds and predicting their properties. Among the earliest descriptors are the classical Zagreb indices introduced by Gutman and Trinajstić in 1972. A more recent development is the hyper-Zagreb index ($HM$), defined as $HM(G)=\sum_{v_i v_j\in E(G)}(d_i+d_j)^2$, where $d_i$ denotes the degree of vertex $v_i$. In 2023, Hayat et al. posed an open problem concerning bounds on the $HM$ index under fixed vertex-connectivity or edge-connectivity, along with the characterization of the corresponding extremal graphs. In this work, the problem is resolved by determining the extremal graphs that maximize $HM$ index under these constraints. The investigation is further extended to several additional extremal problems, including graphs with a given number of leaves, chromatic number, and independence number. The associated extremal graphs are identified in each case. In addition, the chemical relevance of $HM$ is examined through QSPR studies. Finally, the conclusion is presented.

physics.chem-ph↗

arXiv2025

On the Augmented Sombor Index of Graphs

Let $G$ be a connected graph having more than two vertices and let $d_i$ denote the degree of vertex $v_i$ in $G$. Let $E(G)$ represent the edge set of $G$. Then, the augmented Sombor (ASO) index of $G$ is defined as $ASO(G) = \sum_{v_i v_j \in E(G)} \sqrt{(d_i + d_j - 2)^{-1}(d_i^2 + d_j^2)}.$ It is known that the cycle graph $C_n$ uniquely minimizes the ASO index in the class of all $n$-order unicyclic graphs. In this paper, we prove that the unique $n$-order unicyclic graph of maximum degree $n-1$ maximizes the ASO index in the aforementioned unicyclic graph class. We also prove that $ASO(G-v_iv_j)<ASO(G)$ whenever neither of the graphs $G-v_iv_j$ and $G$ contains any isolated edge. Utilizing this edge-deletion property, we characterize the unique graph maximizing the ASO index among all fixed-order connected graphs with a specified vertex connectivity (or edge connectivity).

math.CO↗

arXiv2025

On Hyperbolic Sombor index of graphs

The Hyperbolic Sombor index $HSO(G)$ of a graph $G$ is defined as \begin{align*} HSO(G) = \sum_{v_iv_j \in E(G)} \frac{\sqrt{d_i^{2}+d_j^{2}}}{\min\{d_i,d_j\}}, \end{align*} where $d_i$ and $d_j$ denote the degrees of the vertices $v_i$ and $v_j$, respectively. This index was recently introduced by Barman et al. [Geometric approach to degree-based topological index: Hyperbolic Sombor index, MATCH Commun. Math. Comput. Chem. 95 (2026) 63-94], who explored some of its mathematical properties and applications. However, their work contains several inaccuracies that require correction. In this paper, we first identify and rectify the errors found in the earlier study. We then extend the investigation by establishing new mathematical results for the Hyperbolic Sombor index across various classes of graphs, including trees, unicyclic graphs, and bicyclic graphs. In addition, we derive some lower and upper bounds for $HSO(G)$ in terms of the number of edges, maximum degree and minimum degree, and we characterize the graphs that attain these bounds. Finally, we conclude the paper by outlining potential directions for future research in this emerging area.

math.CO↗

arXiv2025

Resolving Open Problems on the Euler Sombor Index

Recently, the Euler Sombor index $(EUS)$ was introduced as a novel degree-based topological index. For a graph $G$, the Euler Sombor index is defined as $$EUS(G) = \sum_{v_i v_j \in E(G)} \sqrt{d_i^2 + d_j^2 + d_i d_j},$$ where $d_i$ and $d_j$ denote the degrees of the vertices $v_i$ and $v_j$, respectively. Very recently, Khanra and Das \textbf{\bf [Euler Sombor index of trees, unicyclic and chemical graphs, \emph{MATCH Commun. Math. Comput. Chem.} \textbf{94} (2025) 525--548]} proposed several open problems concerning the Euler Sombor index. This paper completely resolves two of the most challenging problems posed therein. First, we determine the minimum value of the $EUS$ index among all unicyclic graphs of a fixed order and prescribed girth, and we characterize the extremal graphs that attain this minimum. Building on this result, we further establish the minimum $EUS$ index within the broader class of connected graphs of the same order and girth, and identify the corresponding extremal structures. In addition, we classify all connected graphs that attain the maximum Euler Sombor index $(EUS)$ when both the order and the number of leaves are fixed.

math.CO↗

arXiv2024

Bounds on the Inverse symmetric division deg index and the relation with other topological indices of graphs

Let $G=(V,E)$ be a simple graph. The concept of Inverse symmetric division deg index $(ISDD)$ was introduced in the chemical graph theory very recently. In spite of this, a few papers have already appeared with this index in the literature. Ghorbani et al. proposed Inverse symmetric division deg index and is defined as $$ISDD(G)=\sum\limits_{v_iv_j\in E(G)}\,\displaystyle{\frac{d_id_j}{d^2_i+d^2_j}},$$ where $d_i$ is the degree of the vertex $v_i$ in $G$. In this paper, we obtain some lower and upper bounds on the inverse symmetric division deg index $(ISDD)$ of graphs in terms of various graph parameters, with identifying extremal graphs. Moreover, we present two relations between the Inverse symmetric division deg index and the various topological indices of graphs. Finally, we give concluding remarks with future work.

math.CO↗

arXiv2024

Unifying adjacency, Laplacian, and signless Laplacian theories

Let $G$ be a simple graph with associated diagonal matrix of vertex degrees $D(G)$, adjacency matrix $A(G)$, Laplacian matrix $L(G)$ and signless Laplacian matrix $Q(G)$. Recently, Nikiforov proposed the family of matrices $A_α(G)$ defined for any real $α\in [0,1]$ as $A_α(G):=α\,D(G)+(1-α)\,A(G)$, and also mentioned that the matrices $A_α(G)$ can underpin a unified theory of $A(G)$ and $Q(G)$. Inspired from the above definition, we introduce the $B_α$-matrix of $G$, $B_α(G):=αA(G)+(1-α)L(G)$ for $α\in [0,1]$. Note that $ L(G)=B_0(G), D(G)=2B_{\frac{1}{2}}(G), Q(G)=3B_{\frac{2}{3}}(G), A(G)=B_1(G)$. In this article, we study several spectral properties of $ B_α$-matrices to unify the theories of adjacency, Laplacian, and signless Laplacian matrices of graphs. In particular, we prove that each eigenvalue of $ B_α(G) $ is continuous on $ α$. Using this, we characterize positive semidefinite $ B_α$-matrices in terms of $α$. As a consequence, we provide an upper bound of the independence number of $ G $. Besides, we establish some bounds for the largest and the smallest eigenvalues of $B_α(G)$. As a result, we obtain a bound for the chromatic number of $G$ and deduce several known results. In addition, we present a Sachs-type result for the characteristic polynomial of a $ B_α$-matrix.

math.CO↗

arXiv2024

On Energy of Graphs with Self-Loops

Let G be a simple graph on n vertices with vertex set V(G). The energy of G, denoted by, $\mathcal{E}(G)$ is the sum of all absolute values of the eigenvalues of the adjacency matrix $A(G)$. It is the first eigenvalue-based topological molecular index and is related to the molecular orbital energy levels of $π$-electrons in conjugated hydrocarbons. Recently, the concept of energy of a graph is extended to a self-loop graph. Let $S$ be a subset of $V(G)$. The graph $G_S$ is obtained from the graph $G$ by attaching a self-loop at each of the vertices of $G$ which are in the set $S$. The energy of the self-loop graph $G_S$, denoted by $\mathcal{E}(G_S)$, is the sum of all absolute eigenvalues of the matrix $A(G_S)$. Two non-isomorphic self-loop graphs are equienergetic if their energies are equal. Akbari et al. (2023)conjectured that there exist a subset $S$ of $V(G)$ such that $\mathcal(G_S) > \mathcal{E}(G)$. In this paper, we confirm this conjecture. Also, we construct pairs of equienergetic self-loop graphs of order 24n for all n \ge 1.

math.CO↗

arXiv2024

Common neighborhood energies and their relations with Zagreb index

In this paper we establish connections between common neighborhood Laplacian and common neighborhood signless Laplacian energies and the first Zagreb index of a graph $\mathcal{G}$. We introduce the concepts of CNL-hyperenergetic and CNSL-hyperenergetic graphs and showed that $\mathcal{G}$ is neither CNL-hyperenergetic nor CNSL-hyperenergetic if $\mathcal{G}$ is a complete bipartite graph. We obtain certain relations between various energies of a graph. Finally, we conclude the paper with several bounds for common neighborhood Laplacian and signless Laplacian energies of a graph.

math.CO↗

arXiv2023

On Zagreb indices of graphs

Let ${\mathcal G}_n$ be the set of class of graphs of order $n$. The first Zagreb index $M_1(G)$ is equal to the sum of squares of the degrees of the vertices, and the second Zagreb index $M_2(G)$ is equal to the sum of the products of the degrees of pairs of adjacent vertices of the underlying molecular graph $G$. The three set of graphs are as follows: \begin{eqnarray*} &&A=\left\{G\in {\mathcal G}_n:\,\frac{M_1(G)}{n}>\frac{M_2(G)}{m}\right\},~B=\left\{G\in {\mathcal G}_n:\,\frac{M_1(G)}{n}=\frac{M_2(G)}{m}\right\} \mbox{ and }&& &&~~~~~~~~~~~~~~~~~~~~~~~~~C=\left\{G\in {\mathcal G}_n:\,\frac{M_1(G)}{n}<\frac{M_2(G)}{m}\right\}. \end{eqnarray*} In this paper we prove that $|A|+|B|<|C|$. Finally, we give a conjecture $|A|<|B|$.

math.CO↗

arXiv2023

Toughness and normalized Laplacian eigenvalues of graphs

Given a connected graph $G$, the toughness $τ_G$ is defined as the minimum value of the ratio $|S|/ω_{G-S}$, where $S$ ranges over all vertex cut sets of $G$, and $ω_{G-S}$ is the number of connected components in the subgraph $G-S$ obtained by deleting all vertices of $S$ from $G$. In this paper, we provide a lower bound for the toughness $τ_G$ in terms of the maximum degree, minimum degree and normalized Laplacian eigenvalues of $G$. This can be viewed as a slight generalization of Brouwer's toughness conjecture, which was confirmed by Gu (2021). Furthermore, we give a characterization of those graphs attaining the two lower bounds regarding toughness and Laplacian eigenvalues provided by Gu and Haemers (2022).

math.CO↗

arXiv2022

On the Sombor index of graphs with given connectivity and number of bridges

Recently in 2021, Gutman introduced the Sombor index of a graph, a novel degree-based topological index. It has been shown that the Sombor index efficiently models the thermodynamic properties of chemical compounds. Assume $\mathbb{B}_n^k$ (resp. $\mathbb{V}_n^k$) comprises all graphs with order $n$ having number of bridges (resp. vertex-connectivity) $k$. Horoldagva & Xu (2021) characterized graphs achieving the maximum Sombor index of graphs in $\mathbb{B}_n^k$. This paper characterizes graphs achieving the minimum Sombor index in $\mathbb{B}_n^k$. Certain auxiliary operation on graphs in $\mathbb{B}_n^k$ are introduced and employed for the characterization. Moreover, we characterize graphs achieving maximum Sombor index in $\mathbb{V}_n^k$. ome open problems, which naturally arise from this work, have been proposed at the end.

math.CO↗

arXiv2022

On distance-regular Cayley graphs of generalized dicyclic groups

Let $G$ be a generalized dicyclic group with identity $1$. An inverse closed subset $S$ of $G\setminus\{1\}$ is called minimal if $\langle S\rangle=G$ and there exists some $s\in S$ such that $\langle S\setminus\{s,s^{-1}\} \rangle\neq G$. In this paper, we characterize distance-regular Cayley graphs $\mathrm{Cay}(G,S)$ of $G$ under the condition that $S$ is minimal.

math.CO↗

arXiv2022

On the spectrum and energy of Seidel matrix for chain graphs

We study various spectral properties of the Seidel matrix $S$ of a connected chain graph. We prove that $-1$ is always an eigenvalue of $S$ and all other eigenvalues of $S$ can have multiplicity at most two. We obtain the multiplicity of the Seidel eigenvalue $-1$, minimum number of distinct eigenvalues, eigenvalue bounds, characteristic polynomial, lower and upper bounds of Seidel energy of a chain graph. It is also shown that the energy bounds obtained here work better than the bounds conjectured by Haemers. We also obtain the minimal Seidel energy for some special chain graphs of order $n$. We also give a number of open problems.

math.CO↗

arXiv2022

Distance-regular Cayley graphs over dicyclic groups

The characterization of distance-regular Cayley graphs originated from the problem of identifying strongly regular Cayley graphs, or equivalently, regular partial difference sets. In this paper, a classification of distance-regular Cayley graphs on dicyclic groups is obtained. More specifically, it is shown that every distance-regular Cayley graph on a dicyclic group is a complete graph, a complete multipartite graph, or a non-antipodal bipartite distance-regular graph with diameter $3$ satisfying some additional conditions.

math.CO↗

arXiv2021

Nordhaus-Guddum type results for the Steiner Gutman index of graphs

Building upon the notion of Gutman index $\operatorname{SGut}(G)$, Mao and Das recently introduced the Steiner Gutman index by incorporating Steiner distance for a connected graph $G$. The \emph{Steiner Gutman $k$-index} $\operatorname{SGut}_k(G)$ of $G$ is defined by $\operatorname{SGut}_k(G)$ $=\sum_{S\subseteq V(G), \ |S|=k}\left(\prod_{v\in S}deg_G(v)\right) d_G(S)$, in which $d_G(S)$ is the Steiner distance of $S$ and $deg_G(v)$ is the degree of $v$ in $G$. In this paper, we derive new sharp upper and lower bounds on $\operatorname{SGut}_k$, and then investigate the Nordhaus-Gaddum-type results for the parameter $\operatorname{SGut}_k$. We obtain sharp upper and lower bounds of $\operatorname{SGut}_k(G)+\operatorname{SGut}_k(\overline{G})$ and $\operatorname{SGut}_k(G)\cdot \operatorname{SGut}_k(\overline{G})$ for a connected graph $G$ of order $n$, $m$ edges and maximum degree $Δ$, minimum degree $δ$.

math.CO↗