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arXiv · 2607.17895

A New Lower Bound on the Spectral Radius of Graphs with Prescribed Average Degree

Sonny Ben-Shimon·Idan Eisner·Shlomo Hoory
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Abstract

This work establishes an improved lower bound for the spectral radius of a graph given its average degree. The new bound follows from an exact solution of the fractional relaxation of the problem. Our findings lead to an affirmative answer to a conjecture by Hong (1993) for graphs with specific average degrees -- as the extremal graphs that meet our bound are proven to have a minimal and maximal degree that differ by at most one. Furthermore, we provide an exact characterization of the conditions that permit such discrete realizations. We prove that for a fixed number of vertices $n$, the number of valid edge configurations grows at least linearly with $n$, achieving an average asymptotic order of $\Theta(n\log n)$.

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Sonny Ben-Shimon, Idan Eisner, Shlomo Hoory. 2026-07-20. A New Lower Bound on the Spectral Radius of Graphs with Prescribed Average Degree. https://arxiv.org/abs/2607.17895

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