Communicated by Michel BrouéKeywords: Irreducible character degree Group order upper bound Let d be the degree of an irreducible character of a finite group G.We can write |G| = d(d +e) for some non-negative integer e. In this document, we prove that if e > 1 then |G| < e 6 − e 4 . This improves an upper bound found by Isaacs of the form Be 6 , where B is an unknown universal constant. We also describe conditions sufficient to sharpen this bound to |G| e 4 − e 3 . In addition, we remove the appeal to the classification of simple groups, which is used in the original paper by Isaacs.

ABSTRACT. Conjecturally, almost all graphs are determined by their spectra. This problem has also been studied for variants such as the spectra of the Laplacian and signless Laplacian. Here we consider the problem of determining graphs with Ihara and Bartholdi zeta functions, which are also computable in polynomial time. These zeta functions are geometrically motivated, but can be viewed as certain generalizations of characteristic polynomials. After discussing some graph properties determined by zeta functions, we show that large classes of cospectral graphs can be distinguished with zeta functions and enumerate graphs distinguished by zeta functions on ≤ 11 vertices. This leads us to conjecture that almost all graphs which are not determined by their spectrum are determined by zeta functions.Along the way, we make some observations about the usual types of spectra and disprove a conjecture of Setyadi and Storm about Ihara zeta functions determining degree sequences.

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