Bilal Ahmad Rather

Bilal Ahmad Rather

The independence polynomial of a graph encapsulates all independent sets of differing sizes, a task classified as NP-hard in theoretical computer science. This article examines the independence polynomial of zero divisor graphs in commutative rings. We demonstrate that the independent sets, represented as a sequence of coefficients of the independence polynomial, exhibit unimodality and log-concavity. Therefore, for the independence polynomial of some zero divisor graphs, the unimodal conjecture is true. Additionally, the characteristics of the zeros of the independence polynomial are delineated, along with their corresponding annular regions on the plane.

Bilal Ahmad Rather

Let $\bigtriangleup$ be a simplicial complex and let $δ_{\mathcal{NF}}$ denote the NF-operator. The NF-complex $δ_{\mathcal{NF}}(\bigtriangleup)$ is defined as the Stanley--Reisner complex of the facet ideal of $\bigtriangleup$. Iterating $δ_{\mathcal{NF}}$ gives a periodic orbit (up to isomorphism), and the smallest positive integer $t$ for which $δ_{\mathcal{NF}}^{\,t}(\bigtriangleup)\cong \bigtriangleup$ is called the \emph{NF-number} of $\bigtriangleup$ (Habi and Mahmood, Algebra Colloquium, 2022). In this work, we provide various results and determine explicit formulas for the NF-number for several families of graphs. In particular, we compute the NF-number for dumbbell graphs. We also prove that the NF-number of the complete split graph $S_{n,m}$ equals $m+n+2$, and that the NF-number of the double star $D_{p+q}$ equals $p+q+4$. We conclude with remarks, open problems, and conjectures to guide future research.

Bilal Ahmad Rather, Mustapha Aouchiche, Victor A. Bovdi

The characteristic polynomials of the Laplacian and the distance Laplacian matrices of power graphs of groups of order $ pqr $, where $ p,q $ and $ r $ are { primes,} are obtained. Further, the characteristic polynomials of these matrices for proper power graphs of cyclic and dicyclic groups are given. The important inequalities for the zeros of the distance Laplacian characteristic polynomials of power graphs of finite groups are presented in comments.

Bilal Ahmad Rather

This article settles Problem 7.2 posed by [Banerjee, Special Matrices (2022)] for the induced subgraph $G_2$ of the comaximal graph $Γ(\mathbb Z_n)$ when $n$ is squarefree. Let $n=p_1p_2\cdots p_m$ with distinct primes $p_1<\cdots<p_m$, and let $G_2$ be the graph on the nonzero nonunit residue classes modulo $n$. We use Chinese remainder representation of $\mathbb Z_n$, and encodes each vertex by the set of vanishing coordinates. This converts $G_2$ into a weighted blow-up of a disjointness graph on nonempty proper subsets of $\{1,\dots,m\}$. Within this model, we derive exact class sizes, explicit degree formulas, the minimum-degree layer, and a short-path criterion. The main theorem proves the connectivity of $G_{2}$ as $κ(G_2)=\prod_{i=1}^{m-1}(p_i-1)=\tfrac{ϕ(n)}{p_m-1}$. Consequently, earlier upper bound is sharp, $G_2$ is maximally connected, and its edge connectivity agrees with its minimum degree. We also obtain distance formulas, diameter and radius information, and a linear-time algorithm once the prime factorization is known.

Bilal Ahmad Rather

For a connected simple graph $G$ on $n$ vertices with chromatic number $χ$, the distance Laplacian matrix is $\DL( G)=\mathrm{diag}(\mathrm{Tr}_{ G}(v_1),\dots,\mathrm{Tr}_{ G}(v_n)) - D( G)$, where $D( G)$ is the distance matrix and $\mathrm{Tr}_{ G}(v)=\sum_{u\in V( G)} d_{ G}(u,v)$ is the transmission. The eigenvalues of $\DL( G)$ are ordered as $\partial^{L}_1( G)\ge \partial^{L}_2( G)\ge \cdots \ge \partial^{L}_n( G)=0$. Building on the chromatic lower bound $\partial^{L}_1( G)\ge n+\ceil{\frac{n}χ}$ and subsequent developments, we prove a \emph{color-class majorization principle}: if $(\ell_1,\dots,\ell_χ)$ are the color-class sizes in an optimal $χ$-coloring with $\ell_1\ge \cdots\ge \ell_χ$, then the first $\ell_1-1$ distance Laplacian eigenvalues satisfy $\partial^{L}_i( G)\ge n+\ell_1$, for $1\le i\le \ell_1-1$. This gives sharp lower bounds on the number of eigenvalues above the chromatic threshold $b_χ=n+\ceil{n/χ}$, thereby refining the distribution theorems of Aouchiche--Hansen (Filomat, 2017) and Pirzada--Khan (LAA, 2021). We further refine clique/independent-set based multiplicity results by deriving explicit chromatic criteria in terms of neighborhood compression, and we generalize the extremal problem for minimum $\partial^{L}_1$ at fixed chromatic number by characterizing all minimizers. Several numerical examples are included along with pictorial representations.

Bilal Ahmad Rather

This study examines the domination polynomials of friendship graphs and book graphs, focusing on unanswered questions related to these families. For the friendship graph $F_n$, with even $n$, we show that the polynomial $D(F_n,x)$ has exactly three real zeros: $0$ and two simple zeros in the intervals $(-2,-1)$ and $(-1,0)$. We further show that these two nonzero zeros have monotonic variation and converge to $-1-\frac{1}{\sqrt2}$ and $-1+\frac{1}{\sqrt2}$, respectively. We obtain the quantitative approximation $(|z|-1)^2\log |z|\le n$ for any complex zeros of $D(F_n,x)$, resulting in the explicit bound $|z|\le 1+\sqrt{\tfrac{n}{\log 2}}$. For book graphs $B_n$, we ascertain the comprehensive limit set of domination roots and establish results about the presence of real roots contingent on parity. We provide a partial answer to the integer-root an issue by establishing that friendship and book graphs have no nonzero integer domination roots, whereas for corona families, the only nonzero integer root is $-2$.

Bilal Ahmad Rather

The general position problem in graphs seeks the largest set of vertices such that no three vertices lie on a common geodesic. Its counting refinement, the general position polynomial $ψ(G)$, asks for all such possible sets. In this paper, We describe general position sets for several classes of graphs and provide explicit formulas for the general position polynomials of complete multipartite graphs. We specialize to balanced complete multipartite graphs and show that for part size $r\le 4$, the polynomial $ψ(K_{r,\dots,r})$ is log-concave and unimodal for all numbers of parts, while for larger $r$, counterexamples show that these properties fail. Finally, we analyze the corona $G\circ K_1$ and prove that unimodality of $ψ(G)$ is retained for numerous natural classes (paths, edgeless graphs, combs). This contributes to an open problem, but the general case remains unknown. Our findings support the parallel between general position polynomials and classical position-type parameters, and identify balanced multipartite graphs and coronas as promising testbeds for additional research.

Bilal Ahmad Rather

We investigate the adjacency matrices of zero-divisor graphs derived from Lipschitz quaternion rings modulo \(n\). For odd primes \(p\), utilizing the isomorphism \(\LL_p\cong M_2(\F_p)\), we categorize vertices by kernel-image type and demonstrate that the adjacency matrix possesses a block structure as a blow-up of a projective incidence matrix. This produces a reduced matrix on the class-constant subspace, with precise formula for the lower bound for the nullity and the multiplicity of the eigenvalue \(-1\), as well as a closed expression for the spectral radius through an equitable partition. For the two-adic family, we precisely ascertain the graph at \(n=2\) and demonstrate that for \(t\ge 2\), the graph \(G_{2^t}\) encompasses substantial cliques derived from the ideal filtering, which yield definitive lower bounds for the spectral radius. We also examine the implications for graph energy and provide a systematic construction of the adjacency matrix.

Bilal Ahmad Rather

This paper studies the zero-divisor graphs attached to several finite chain-ring families and computes the homological invariants of their edge ideals by using cochordal constructible systems. We begin with a general layered graph $C(q,L)$, whose vertices are arranged according to valuation layers and whose adjacency is governed by the single rule $k+\ell\ge L$, form some integers $k$ and $\ell$. This graph models the zero-divisor structure of a finite chain ring with residue field of order $q$ and nilpotency index $L$. We prove that $C(q,L)$ is cochordal, determine its type sequence, then correct and refine the Betti formula of its edge ideal [Dung and Vu, Cochordal zero divisor graphs and Betti numbers of their edge ideals, Comm. Algebra 54(2) (2026) 736--744]. The results are then specialized to the Gaussian quotient rings $\mathbb Z_{2^m}[i]$ and to the truncated polynomial rings $\mathbb Z_p[x]/(x^c)$. We compute projective dimension, regularity, independence number, height, Hilbert series, and Cohen--Macaulay behavior. The computations show that these quotient rings have $2$-linear resolutions, while Cohen--Macaulayness occurs only in the expected degenerate or complete-graph cases.

Bilal Ahmad Rather

The comaximal graph $ Γ(R) $ of a commutative ring $R$ is a simple graph with vertex set $ R $ and two distinct vertices $ a $ and $b $ of $ Γ(R) $ are adjacent if and only if $ aR+bR=R $, where $ aR $ is the ideal generated by $ a $ in $ R $. In this article, the independent domination polynomial $ D_{i}(Γ(\mathbb{Z}_{n}),x) $ of $ Γ(\mathbb{Z}_{n}) $ is discussed, along with its unimodal and log-concave properties for certain values of $n$. Some auxiliary results related to $D_{i}(Γ(\mathbb{Z}_{n}),x)$ are presented in terms of their zeros. In addition, we determine the independence polynomial $ I(Γ(\mathbb{Z}_{n}),x ) $ of $ Γ(\mathbb{Z}_{n}) $ for special values of $n$ and provide a general result associated with it. The bounds for the zero of the polynomial $ I(Γ(\mathbb{Z}_{n}),x ) $ are established, and their log-concave and unimodal properties are examined.

Bilal Ahmad Rather

Let $R$ be a commutative ring with identity and let $Z^{\ast}(R)$ denote the set of nonzero zero-divisors of $R$. The \emph{zero-divisor graph} $ \varGamma(R)$ is the simple graph with vertex set $V( \varGamma(R))=Z^{\ast}(R)$, where two distinct vertices$x,y\in Z^{\ast}(R)$ are adjacent if and only if $xy=0$ in $R$. In this paper we investigate the zero-divisor graph of the truncated polynomial ring $R=\mathbb{Z}_{p}[x]/\langle x^{c}\rangle,$ for $c\in\mathbb{N}.$ We determine the spectrum of the $A_α$-matrix associated with $ \varGamma(R)$, and, as special cases, explicitly obtain both the adjacency spectrum and the signless Laplacian spectrum of $ \varGamma(R)$. Furthermore, we prove that the Laplacian eigenvalues, as well as the distance eigenvalues, of these graphs are all integers.