B. Rather

activity

20202026

most citedOn

α

-adjacency energy of graphs and Zagreb index

8 citations · 21 across the 25 of their papers we have counts for

collaborators

26 papers

math.CO2026

Metric Bases of Barycentric and Matching Subdivisions of Zero-Divisor Graphs

Vidya S, Prasanna Poojary, Bilal Ahmad Rather +1

In this paper, we study metric bases and related metric properties for barycentric and partial matching subdivisions of the zero-divisor graph of $Z_{pq}$ , where $p$ and $q…

math.AC2026

Homological invariants of edge ideals of the multiple extended complete split-like graphs

Bilal Ahmad Rather

We study the graphs $MECS_{b,n}^a \cong \overline{K}_a \join \big(n(K_b+K_2)\big)$, obtained by attaching an independent set of size $a$ to $n$ disjoint copies of the block $K_b+K_…

math.CO2026

Graded Betti numbers of generalized split--join graphs and applications

Bilal Ahmad Rather

We determine the full graded Betti tables of graph families that subsume several classes studied recently in the literature, namely the generalized multiple complete split-like gra…

math.CO2026

Combinatorial and analytic aspects of independence polynomials of zero divisor graphs

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…

math.CO2026

The NF-operator and the NF-Numbers of Simplicial Complexes

Bilal Ahmad Rather

Let $△$ be a simplicial complex and let $δ_{N F}$ denote the NF-operator. The NF-complex $δ_{N F} (△)$ is defined as the Stanley--Reisner…

math.CO2026

Extremal chromatic bounds for distance Laplacian eigenvalues

Bilal Ahmad Rather

For a connected simple graph $G$ on $n$ vertices with chromatic number $χ$ , the distance Laplacian matrix is $\DL(G)=\operatorname{diag}(\Tr_G(v_1),\dots,\Tr_G(v_n))-D(G)$, where $…

math.AC2026

Betti numbers for cochordal zero-divisor graphs of commutative rings

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 constructi…

math.CO2026

Vertex connectivity of the nonzero nonunit core of the comaximal graph of $Z_{n}$

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 $Γ (Z_{n})$ when $n$ is squarefree. Let $n=…

math.CO2026

On roots of domination polynomials for friendship and book graphs

Bilal Ahmad Rather

This study examines the domination polynomials of friendship graphs and book graphs, focusing on unanswered questions related to these families [Alikhani, Brown and Jahari, on the…

math.CO2026★ 5 cited

On (distance) Laplacian characteristic polynomials of power graphs

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 $pq r$ , where $p, q$ and $r$ are { primes,} are obtained.…

math.CO2026

Independent domination polynomial of comaximal graphs of commutative rings

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…

math.CO2026

On the quotient matrix and its distinct eigenvalues

Bilal Ahmad Rather, Jianfeng Wang, Zhaoxi Li

Let $M$ be an $n \times n$ matrix partitioned into blocks according to a partition $P = {C_{1}, \dots, C_{ℓ}}$ of its index set, and let $Q$ be the corresponding equitable quotient…

math.CO2026

Spectral Properties of Zero-Divisor Graphs of Truncated Polynomial Rings

Bilal Ahmad Rather

Let $R$ be a commutative ring with identity and let $Z^{*} (R)$ denote the set of nonzero zero-divisors of $R$ . The \emph{zero-divisor graph} $Γ (R)$ is the simple graph…

math.CO2026

Block Structure and Spectrum of Zero-Divisor Graphs of Lipschitz Quaternion Rings Modulo $n$

Bilal Ahmad Rather

We study the adjacency spectra of zero-divisor graphs associated with Lipschitz quaternion rings modulo $n$ . For an odd prime $p$ , the identification $\mathbb{L}_p\cong M_2(\mathbb…

math.CO2026

Domination polynomial of co-maximal graphs of integer modulo ring

Bilal Ahmad Rather

We investigate the domination polynomial of the co-maximal graph $Γ (Z_{n})$ related to the ring of integers modulo $n$ . Explicit formulas are derived for $ n = p^{n_1} $ a…

math.CO2026

Degree-based weighted adjacency matrices: spectra, integrality, and edge deletion effects

Bilal Ahmad Rather, Hilal Ahmad Ganie

We study degree-based weighted adjacency matrices associated with symmetric edge-weight functions $ϕ (d_{u}, d_{v})$ , with emphasis on complete multipartite graphs and spectral changes c…

math.CO2026

Explicit Formulas and Unimodality Phenomena for General Position Polynomials

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 polynom…

math.CO2024★ 1 cited

A note on eigenvalues of zero divisor graphs associated with commutative rings

Bilal Ahmad Rather

For a commutative ring $R,$ with non-zero zero divisors $Z^{*} (R)$ . The zero divisor graph $Γ (R)$ is a simple graph with vertex set $Z^{*} (R)$ , and two distinct vertices $x,y…

math.CO2024★ 1 cited

A note on the Independent domination polynomial of zero divisor graph of rings

Bilal Ahmad Rather

In this note we consider the independent domination polynomial problem along with their unimodal and log-concave properties which were earlier studied by Gürsoy, Ülker and Gürsoy (…

math.CO2023

Proximity and Remoteness in Graphs: a survey

Mustapha Aouchiche, Bilal Ahmad Rather

The proximity $π = π (G)$ of a connected graph $G$ is the minimum, over all vertices, of the average distance from a vertex to all others. Similarly, the maximum is called the remote…

math.CO2021

On distance Laplacian energy in terms of graph invariants

Hilal A. Ganie, Rezwan Ul Shaban, Bilal A. Rather +1

For a simple connected graph $G$ of order $n$ having distance Laplacian eigenvalues $ρ_{1}^{L} \geq ρ_{2}^{L} \geq \dots \geq ρ_{n}^{L}$ , the distance Laplacian energy $D L E (G)$ is…

math.CO2021★ 2 cited

On $A_{α}$ -spectrum of joined union of graphs and its applications to power graphs of finite groups

Bilal A. Rather, Hilal A. Ganie, S. Pirzada

For a simple graph $G$ , the generalized adjacency matrix $A_{α} (G)$ is defined as $A_{α} (G) = α D (G) + (1 - α) A (G), α \in [0, 1]$ , where $A (G)$ is the adjacency matrix and $D (G)$ is the diagona…

math.CO2021

On a conjecture of Laplacian energy of trees

Hilal A. Ganiea, Bilal A. Rather, S. Pirzada

Let $G$ be a simple graph with $n$ vertices, $m$ edges having Laplacian eigenvalues $μ_{1}, μ_{2}, \dots, μ_{n - 1}, μ_{n} = 0$ . The Laplacian energy $L E (G)$ is defined as $LE(G)=\sum_{i=1}^{…

math.CO2021★ 4 cited

On Distribution of Laplacian Eigenvalues of Graphs

Bilal A. Rather

The work in this thesis concerns the investigation of eigenvalues of the Laplacian matrix, normalized Laplacian matrix, signless Laplacian matrix and distance signless Laplacian ma…

Metric Bases of Barycentric and Matching Subdivisions of Zero-Divisor Graphs

Homological invariants of edge ideals of the multiple extended complete split-like graphs

Graded Betti numbers of generalized split--join graphs and applications

Combinatorial and analytic aspects of independence polynomials of zero divisor graphs

The NF-operator and the NF-Numbers of Simplicial Complexes

Extremal chromatic bounds for distance Laplacian eigenvalues

Betti numbers for cochordal zero-divisor graphs of commutative rings

Vertex connectivity of the nonzero nonunit core of the comaximal graph of Zn​

On roots of domination polynomials for friendship and book graphs

On (distance) Laplacian characteristic polynomials of power graphs

Independent domination polynomial of comaximal graphs of commutative rings

On the quotient matrix and its distinct eigenvalues

Spectral Properties of Zero-Divisor Graphs of Truncated Polynomial Rings

Block Structure and Spectrum of Zero-Divisor Graphs of Lipschitz Quaternion Rings Modulo n

Domination polynomial of co-maximal graphs of integer modulo ring

Degree-based weighted adjacency matrices: spectra, integrality, and edge deletion effects

Explicit Formulas and Unimodality Phenomena for General Position Polynomials

A note on eigenvalues of zero divisor graphs associated with commutative rings

A note on the Independent domination polynomial of zero divisor graph of rings

Proximity and Remoteness in Graphs: a survey

On distance Laplacian energy in terms of graph invariants

On Aα​-spectrum of joined union of graphs and its applications to power graphs of finite groups

On a conjecture of Laplacian energy of trees

On Distribution of Laplacian Eigenvalues of Graphs

Vertex connectivity of the nonzero nonunit core of the comaximal graph of $Z_{n}$

Block Structure and Spectrum of Zero-Divisor Graphs of Lipschitz Quaternion Rings Modulo $n$

On $A_{α}$ -spectrum of joined union of graphs and its applications to power graphs of finite groups