arXiv subjects

Priti Prasanna Mondal

Publications and source records attributed to Priti Prasanna Mondal.

5 recordsLinked to original sources

arXiv2026

Spectral Radius, Vertex Deletion, and Chromatic Number of Signed Graphs

A signed graph $Σ=(G,σ)$ is a graph $G$ with edges given signs $1$ or $-1$ defined by the function $σ$. The adjacency matrix of $Σ$ is defined as per these signs. The relation between the largest eigenvalue of $G$ and $G-v$ has been studied in recent years, where $G-v$ is the graph obtained from $G$ by deleting the vertex $v$. In 2020, Sun and Das proved that the difference of the squares of the largest eigenvalues of the graphs $G$ and $G-v$ is bounded above by $2d(v)-1$ where $d(v)$ is the degree of $v$. A similar result need not be true for the largest eigenvalue of signed graphs. In this paper, we prove that the result is valid for the spectral radius of signed graphs. On the other hand, the signed graph version of Hoffman's chromatic number bound was proved by Wang et al. in 2021. They also discussed the difficulty in proving the extended version encompassing all eigenvalues of $Σ$ as was done for unsigned graphs by Wocjan and Elphick. We note down a consequence of Wocjan and Elphick's lower bound for the chromatic number in terms of all the eigenvalues of $Σ_+$; all the eigenvalues of $Σ$ and $Σ_-$, where $Σ_+$ (resp. $Σ_-$) is the spanning subgraph induced by the positive (resp. negative) edges. We give examples where the result fails even under various restrictions on the signed graph. Finally, we improve an upper bound for the $k$-th power of the largest eigenvalue given by Stanić in terms of walks in signed graphs and give lower bounds for the least eigenvalue in terms of various parameters of $Σ_+$ and $Σ_-$.

math.CO↗

arXiv2026

Spectral radius and second largest eigenvalues of power graphs of finite groups

Consider a group $\mathbb{G}$ and construct its power graph, whose vertex set consists of the elements of $\mathbb{G}$. Two distinct vertices (elements) are adjacent in the graph if and only if one element can be expressed as an integral power of the other. In this article, we improved the bounds of the spectral radius of the power graphs of the cyclic group $C_{n}$, the dihedral group $\mathcal{D}_{2n}$, and the dicyclic group $\mathcal{Q}_{4n}$. For $n\neq p^{m},$ the power graph of the cyclic group $C_{n}$ is not a complete multipartite graph. We find the second largest eigenvalue bounds of the same with the clique number. In some cases, we find the bounds are exact if and only if they belong to a particular family of graphs. Lastly, we work on the distance spectral radius of the power graphs of the same groups

math.SP↗

arXiv2025

When Are Standard Graph Products Isomorphic?

This article investigates the isomorphism problem for graphs derived from the four standard graph products: Cartesian, Kronecker (direct), strong, and lexicographic product. We provide a complete characterization of all simple connected graphs for which their corresponding products are isomorphic. As a by-product, we identify a novel family of non-distance-regular graphs that possess fewer than d+1 distinct distance eigenvalues, where d represents the diameter of the graph. This result offers a new perspective on Problem 4.3 posed in [2], moving beyond the current approaches.

math.CO↗

arXiv2025

On the Spectral Analysis of Power Graph of Dihedral Groups

The power graph $ \mathcal{G}_G $ of a group $ G $ is a graph whose vertex set is $ G $, and two elements $ x, y \in G $ are adjacent if one is an integral power of the other. In this paper, we determine the adjacency, Laplacian, and signless Laplacian spectra of the power graph of the dihedral group $ D_{2pq} $, where $ p $ and $ q $ are distinct primes. Our findings demonstrate that the results of Romdhini et al. [2024], published in the \textit{European Journal of Pure and Applied Mathematics}, do not hold universally for all $ n \geq 3 $. Our analysis demonstrates that their results hold true exclusively when $ n = p^m $ where $ p $ is a prime number and $ m $ is a positive integer. The research examines their methodology via explicit counterexamples to expose its boundaries and establish corrected results. This study improves past research by expanding the spectrum evaluation of power graphs linked to dihedral groups.

math.SP↗

arXiv2024

On the distance spectrum of the Kronecker product of distance regular graphs

Consider two simple graphs, G1 and G2, with their respective vertex sets V(G1) and V(G2). The Kronecker product forms a new graph with a vertex set V(G1) X V(G2). In this new graph, two vertices, (x, y) and (u, v), are adjacent if and only if xu is an edge in G1 and yv is an edge in G2. While the adjacency spectrum of this product is known, the distance spectrum remains unexplored. This article determines the distance spectrum of the Kronecker product for a few families of distance regular graphs. We find the exact polynomial, which expresses the distance matrix D as a polynomial of the adjacency matrix, for two distance regular graphs, Johnson and Hamming graphs. Additionally, we present families of distance integral graphs, shedding light on a previously posted open problem given by Indulal and Balakrishnan in (AKCE International Journal of Graphs and Combinatorics, 13(3); 230 to 234, 2016).

math.SP↗