Research and Projects

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My research is centered around spectral graph theory, where I study graphs through the eigenvalues of matrices associated with them. A central theme in my work is to understand how spectral information constrains the structure of a graph, especially when the eigenvalues are restricted in some way.

I am particularly interested in problems at the interface of spectral graph theory, extremal combinatorics, and discrete geometry. In my work on graphs with restricted smallest eigenvalue, spectral classification problems connect naturally to root systems and to questions motivated by spherical two-distance sets. In other projects, extremal examples arise from incidence structures such as projective planes. I am interested in understanding how these geometric and combinatorial structures interact with graph spectra.

More recently, I have also become interested in topological combinatorics, especially in using Borsuk-Ulam type results and related topological ideas to study discrete problems.

Broadly, I am interested in problems where algebraic, geometric, and topological methods interact with combinatorics to uncover structure in discrete objects.

Research Papers

Bounds on median eigenvalues of graphs of bounded degree (submitted)

Hricha Acharya, Zilin Jiang, Shengtong Zhang

We prove that for every integer d ≥ 3, the median eigenvalues of any graph of maximum degree d are bounded above by √ d − 1. We also prove that, in three separate cases, the median eigenvalues of a graph of maximum degree d are bounded below by − √ d − 1: when the graph is triangle-free, when d − 1 is a perfect square, or when d ≥ 75. These results resolve, for all but finitely many values of d, an open problem of Mohar on median eigenvalues of graphs of maximum degree d. As a byproduct, we establish an upper bound on the average energy of graphs of maximum degree at most d, generalizing a previous result of van Dam, Haemers, and Koolen for d-regular graphs.

Median eigenvalues of subcubic graphs (under review)

Hricha Acharya, Bejamin Jeter, Zilin Jiang

In mathematical chemistry, graphs where each vertex connects to at most three others are often referred to as chemical graphs. In the Hückel molecular orbital model, the median eigenvalues of these graphs correspond to the highest occupied and lowest unoccupied molecular orbitals (HOMO and LUMO), which are crucial for understanding a molecule's electronic properties. In 2010, Fowler and Pisanski conjectured that, with only a finite number of exceptions, the median eigenvalues of chemical graphs fall within the interval [−1, 1]. They confirmed this for all chemical trees. Building upon this, Mohar in 2013 extended the validation to all bipartite planar chemical graphs and, later in 2016, to all bipartite chemical graphs except the Heawood graph. We fully resolve this conjecture by proving that, except the Heawood graph, the median eigenvalues of all connected chemical graphs lie within [−1, 1]. Additionally, we demonstrate that a positive fraction of the eigenvalues around the median eigenvalues also reside in this interval, mirroring Mohar's findings for bipartite chemical graphs.

Beyond the classification theorem of Cameron, Goethals, Seidel, and Shult

Hricha Acharya, Zilin Jiang.

Combinatorics, Probability and Computing · 2026 · DOI

A central problem in spectral graph theory is to characterize graphs with bounded adjacency eigenvalues. It is well-known that for the infinite family of line graphs, their adjacency eigenvalues are always greater than or equal to -2. In 1976, Cameron, Goethals, Seidel, and Shult gave a complete characterization of graphs whose smallest eigenvalue is at least -2. Later, work by Hoffman (1972) and Jiang and Polyanskii (2021) showed that x*, the smallest value greater than 2, for which there are infinitely many graphs with the smallest eigenvalue at least -x, is approximately 2.0198. This raises intriguing questions: What makes x* special? What characteristics do graphs with smallest eigenvalues in the range (-x*, -2) exhibit? In our research, we addressed these questions and provided a complete characterization of graphs whose smallest eigenvalue is in the range (-x*,-2). Our result is the first classification of infinitely many connected graphs with their smallest eigenvalue in the interval (-x, -2) for any constant x > 2.

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