arXiv subjects

Sauvik Poddar

Publications and source records attributed to Sauvik Poddar.

6 recordsLinked to original sources

arXiv2026

Cayley colour integral groups

A finite group $G$ is said to be Cayley integral if every undirected Cayley graph $\operatorname{Cay}(G,S)$ on $G$ is integral. In this paper, we introduce three natural extensions of this concept; namely as: Cayley colour integral, $\mathfrak{F}$-Cayley colour integral and normal Cayley integral groups. We characterize the first two families in its entirety. The last family of groups is shown to be coinciding with inverse semi-rational groups introduced by Chillag and Dolfi, thereby providing an alternative characterization for the same. We also establish an inclusion hierarchy among these families.

math.CO↗

arXiv2026

Homogenized Graphical Shi Arrangements and Deformed Dumont Permutations

We introduce the homogenized graphical Shi arrangement associated with a simple undirected graph $G$, which serves as a broad generalization of classical deformations of the braid arrangement, including the Shi and homogenized Linial arrangements studied by Lazar and Wachs. We demonstrate that the intersection lattices of certain homogenized graphical Shi arrangements are isomorphic to the bond lattices of naturally associated graphs. For a distinguished family of graphs, by employing non-broken-circuit (NBC) techniques, we obtain explicit combinatorial interpretations of the arrangement's M\"obius function and characteristic polynomial. We achieve this by introducing generalisations of previously studied combinatorial objects: $\mathcal{R}$-deformed increasing-decreasing ($\mathcal{R}$-DID) forests and $\mathcal{R}$-D-permutations, establishing bijections between them to interpret the coefficients of the characteristic polynomial in terms of $\mathcal{R}$-D-permutations with prescribed number of cycles. Furthermore, we explore refinements of $\mathcal{R}$-D-permutations by their starting letters. Finally, we also resolve an open conjecture posed by Deutsch, Kitaev, and Remmel concerning the equidistribution of specific parity-constrained descent and ascent statistics.

math.CO↗

arXiv2026

Algebraic degree of Cayley colour graphs

The splitting field of a graph $\Gamma$ with respect to a square matrix $M$ associated with $\Gamma$, is the smallest field extension over the field of rationals $\mathbb{Q}$ that contains all the eigenvalues of $M$. The degree of the extension is called the algebraic degree of $\Gamma$ with respect to $M$. In this paper, we completely determine the splitting field of the adjacency matrix of the Cayley colour graph $\operatorname{Cay}(G,f)$ on a finite group $G$, associated with a class function $f:G\to\mathbb{Q}$ and compute its algebraic degree, which generalize the main results of Wu et al. Moreover, we study the relation between the algebraic integrality of two Cayley colour graphs, and deduce the fact that the algebraic degree and distance algebraic degree of a normal Cayley graph are same, generalizing a result of Zhang et al.

math.CO↗

arXiv2025

An integral family of quasi-strongly regular Cayley graphs

Quasi-strongly regular graphs form a significant generalization of strongly regular graphs. We study the eigenvalues of a family of such graphs, $\Gamma_H(G)$, constructed from a finite group $G$ and a subgroup $H$. Our main results include a sufficient condition for $\Gamma_H(G)$ to be integral and an explicit computation of its entire spectrum when $H$ is normal, revealing that the spectrum in this case depends only on $|G|$ and the index $[G:H]$.

math.CO↗

arXiv2025

Prime Order Element Graph of a Group -- II

In this sequel paper, we continue the analysis of the prime order element graph $\Gamma(G)$ of a finite group $G$, where vertices are elements of $G$ and edges connect distinct elements $x, y$ satisfying $\circ(xy) = p$ for some prime $p$. Our investigation focuses on the adjacency and Laplacian spectra, planarity, and clique number of this graph. We conclude by outlining open issues and potential directions for future investigations.

math.GR↗

arXiv2025

Non-isomorphic $d$-integral circulant graphs

The algebraic degree $Deg(G)$ of a graph $G$ is the dimension of the splitting field of the adjacency polynomial of $G$ over the field $\mathbb{Q}$. It can be shown that for every positive integer $d$, there exists a circulant graph with algebraic degree $d$. Let $C(d)$ be the least positive integer such that there exists a circulant graph of order $C(d)$ having algebraic degree $d$. A graph $G$ is called $d$-integral if $Deg(G)=d$. We call a $d$-integral circulant graph \textit{minimal} if order of that graph equals $C(d)$. Let $\mathcal{F}_{n,d}$ denote the collection of isomorphism classes of connected, $d$-integral circulant graphs of some given possible order $n$. In this paper we compute the exact value of $C(d)$ and provide some bounds on $|\mathcal{F}_{n,d}|$, thereby showing that the minimal $d$-integral circulant graph is not unique. Moreover, we find the exact value of $|\mathcal{F}_{p,d}|$ where both $p$ and $d$ are prime.

math.CO↗