Bilal earned his Ph.D, PGDCA, B.Ed, M.Sc, and B.Sc. from the University of Kashmir. He worked as a postdoctoral fellow at UAE University for three years. His research interests include spectral properties of graph matrices, combinatorial commutative algebra, algebraic combinatorics, and theoretical computer science. Currently, He is an Assistant Professor at SIUT, Uzbekistan, focusing on teaching and research, with a primary interest in zeros of graph polynomials and Quaternion polynomials, homological invariants of edge ideals and their homotopy type, codes on paley graphs, matrix operators
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Education
Ph.D Mathematics, 11th May 2017 to 15-th February (2021), University of Kashmir, Srinagar Kashmir, India.
Thesis Title: On Distribution of Laplacian Eigenvalues of Graphs.
Post Graduate Diploma in Computer Applications (2015-2017), University of Kashmir, Kashmir, India.
Dissertation: Solution of various problems in numerical analysis using C programming
Bachelor of Education (Mathematics Teaching, Principles of Education, Psychology of Learning) (2013-2015), University of Kashmir, Srinagar Kashmir, India.
Project: Evaluation of Mathematics teaching at school level.
M.Sc. Mathematics (2011-2013), University of Kashmir, Srinagar Kashmir, India.
Major Courses: Real Analysis I and II, Abstract Algebra I and II, Functional Analysis I and II, General Topology, Linear Algebra, Numerical Analysis, Mathematical Statistics I and II, Measure Theory, Programming, Graph Theory, Mathematical Modelling, Differential and Partial differential equations, Number theory I and II, Complex Analysis I and II, Analytic Polynomial theory, Differential Geometry.
Dissertation: Alternative Proof and Applications of Arithmetic-Geometric Mean Inequality.
Published in 11th JK Science Congress Proceeding 2015, University of Kashmir, Kashmir (Oct 12-14, 2015), ISBN 987-93-82288-87-9, P. 269--273.
B.Sc. with Electronics, Physics, Mathematics, (2008-2010), University of Kashmir, Srinagar Kashmir, India.
JKSET (Jammu and Kashmir State Eligibility Test for Assistant Professorship) (2018).
Erdős Number=3 (Rather, Bilal Ahmad-->Gutman, Ivan-->Clark, Lane H.)
Awards/Nominations and Recognitions
Post doctoral fellowships
Department of Mathematics, Indian Institute of Technology, Guwahati, India, 2024 (Prof Sukanta Pati, not joined).
Department of Mathematics and Statistics, Shandong University of Science and Technology, Zibo, China, 2023 (Prof Jainfeng Wang), awarded 10 year talent R visa.
Mathematical Sciences Department, United Arab Emirates University, Al Ain 15551, Abu Dhabi, UAE, 2021 (Dr Mustapha Aouchiche and Dr Muhammad Imran).
Department of Mathematics, University of Sharjah, Sharjah, UAE 2023 (Prof Zahid Raza, not joined).
Awarded ``Industrial-Young Scientist Award" by Computer Vision Awards (2024), see https://computer-vision-conferences.scifat.com/bilal-ahmad-rather-industrial-young-scientist-award-6942/
Nominated for Annual Research Awards 2021 by Dr K. V. Rao Scientific Society (KVRSS) Hyderabad, for the paper entitled ``Distance signless Laplacian spectrum of graphs and spectrum of zero divisor graphs of $\mathbb{Z}_n$, Linear and Multilinear Algebra 70(17) (2022) 3354-3369"
JKSET (state eligibility test for Assistant Professorship/Lectureship, UGC India), 2018.
Junior research fellowship for a period of two years on SERB-DST(EMR/ 2015/0010471) major research project with the principal investigator Prof Sharieffudin Pirzada, entitled ``On energy and Laplacian energy of graphs'' at Department of Mathematics, University of Kashmir, Srinagar Kashmir, India.
Major Research Contributions
Settled Brouwer's Conjecture for bi-regular graphs, graphs with c-cyclic subgraphs and other families of graphs.
Proved Laplacian energy conjecture (trace norm of Laplacian matrix) for trees of diameter $4$ and beyond.
Solved conjecture for normalized distance matrix with two distinct eigenvalues.
Disproved Grešgorin discs conjecture about the distance matrices of graphs.
Solved an open problem related to distinct eigenvalues of quotient matrices.
Solved several open problems related to integrability of graphs with various matrices associated to them.
Research Interests
Algebraic graph theory especially spectral theory of graphs, their applications in Linear Algebra and Abstract Algebra: Eigenvalue and eigenvector properties of Adjacency matrix, Laplacian matrix, Distance matrix, and spectral analysis of power graphs and commuting graphs of finite groups and zero-divisor graphs of finite commutative rings.
Operator norms and their applications in spectral theory of graphs.
Betti numbers, projective dimension, regularity of the linear strand of edge ideals, Homotopy type, Hilbert Series.
Graph polynomials like Independent, domination and independent domination polynomial of graphs their properties: log concavity, unimodality and location of their zeros. Applications of domination, independent, independent domination theory in theoretical computer science and coding theory.
Representative Publications:
Bilal Ahmad Rather, J. Wang and F. Balarado, Independent domination polynomial of binary sequence graphs, Journal of Algebraic Combinatorics 63 (2026), Article number 57, https://doi.org/10.1007/s10801-026-01514-x.
Bilal Ahmad Rather, Hilal A. Ganie and Y. Shang, Distance Laplacian eigenvalues of sun graphs, Applied Mathematics and Computations 445 (2023), 127847, https://doi.org/10.1016/j.amc.2023.127847.
Bilal Ahmad Rather, X. Gao and J. Wang, Spectral spread and energy of eccentricity matrix, Discrete Applied Mathematics 352 (2024) 9--19, https://doi.org/10.1016/j.dam.2024.03.020.
Bilal Ahmad Rather, Complex Zeros and Log-Concavity in Independent Domination Polynomials of Zero Divisor Graphs of Commutative Rings, Theoretical Computer Science 1058 (2025) Article number 115594, https://doi.org/10.1016/j.tcs.2025.115594.
Bilal Ahmad Rather, Fiber decompositions and new invariants for zero-divisor graphs induced by unfaithful modules, Journal of Algebra and its Applications, (2026), https://doi.org/10.1142/S021949882850048X.
Bilal Ahmad Rather, M. Imran and A. Diene, The linear strand of edge ideals of comaximal graphs of commutative rings, Communications in Algebra 52(4) (2024) 1486--1500, https://doi.org/10.1080/00927872.2023.2263559.
Bilal Ahmad Rather, Independent domination polynomial of comaximal graphs of commutative rings, Algebra colloquium, 33(2) (2026) 243--258, https://doi.org/10.1142/S1005386726000222.
M. Aouchiche, Bilal Ahmad Rather and I. El Hallaoui, On the Grešgorin discs of distance matrices of graphs, Electronic Journal of Linear Algebra 37 (2021) 709--717, http://doi.org/10.13001/ela.2021.6489.
S. Pirzada, Bilal Ahmad Rather, M. Aijaz and T. A. Chishti, Distance signless Laplacian spectrum of graphs and spectrum of zero divisor graphs of Z_n, Linear and Multilinear Algebra 70(17) (2022) 3354--3369, http://doi.org/10.1080/03081087.2020.1838425.
Hilal A. Ganie, S. Pirzada, Bilal Ahmad Rather and V. Trevisan, Further developments on Brouwer's conjecture for the sum of Laplacian eigenvalues of graphs, Linear Algebra and its Applications 588 (2020) 1--18. https://doi.org/10.1016/j.laa.2019.11.020.
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