2026/07/20 by Sonny Ben-Shimon, Idan Eisner, Shlomo Hoory
Mathematics · #math.CO
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 Θ(nlog n).