arXiv subjects

Hricha Acharya

Publications and source records attributed to Hricha Acharya.

3 recordsLinked to original sources

arXiv2026

Bounds on median eigenvalues of graphs of bounded degree

We prove that for every integer $d \ge 3$, the median eigenvalues of any graph of maximum degree $d$ are bounded above by $\sqrt{d-1}$. We also prove that, in three separate cases, the median eigenvalues of a graph of maximum degree $d$ are bounded below by $-\sqrt{d-1}$: when the graph is triangle-free, when $d-1$ is a perfect square, or when $d \ge 75$. 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.

math.CO↗

arXiv2025

Median eigenvalues of subcubic graphs

We show that the median eigenvalues of every connected graph of maximum degree at most three, except for the Heawood graph, are at most $1$ in absolute value, resolving open problems posed by Fowler and Pisanski, and by Mohar.

math.CO↗

arXiv2024

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

In 1976, Cameron, Goethals, Seidel, and Shult classified all the graphs whose smallest eigenvalue is at least $-2$ by relating such graphs to root systems that appear in the classification of semisimple Lie algebras. In this paper, extending their beautiful theorem, we give a complete classification of all connected graphs whose smallest eigenvalue lies in $(-\lambda^*, -2)$, where $\lambda^* = \rho^{1/2} + \rho^{-1/2} \approx 2.01980$, and $\rho$ is the unique real root of $x^3 = x + 1$. Our result is the first classification of infinitely many connected graphs with their smallest eigenvalue in $(-\lambda, -2)$ for any constant $\lambda > 2$.

math.CO↗