In this paper, we investigate the eigenvalues of character degree graphs, with particular emphasis on the arithmetic properties of their spectra. First, we study \((n-2)\)-regular character degree graphs of solvable groups and derive an explicit formula for their characteristic polynomials. We show that all their eigenvalues are rational and, consequently, that their splitting field is \(\mathbb{Q}\). We then consider supergraphs obtained by adding edges to these graphs and prove that the corresponding splitting field is a quadratic extension of \(\mathbb{Q}\). Next, using their structural decomposition, we examine a general class of Lewis graphs. For this class, we establish bounds on both the number of irrational eigenvalues and the degree of the associated splitting fields. Finally, we investigate prime character degree graphs of diameter \(3\), focusing on the arithmetic nature of their eigenvalues and the degree of their splitting fields.
This thesis investigates two central directions in algebraic graph theory, with an emphasis on spectral methods: spectral determination of graphs and transitivity properties of generalized-Hamming graphs and their complements. The first part focuses on graphs that are determined by the spectra of associated matrices. W...
In this paper, we resolve a 30-year-old conjecture of Spielman and Teng concerning the performance of the spectral partitioning method on graphs embeddable on an orientable surface of genus $g\ge 1$. In particular, for such a graph $G$ with $n$ vertices and maximum degree $\Delta$, we show that the second-smallest eige...
Benedikt Kolbe, Jack Spalding-Jamieson· 0 citations
An eigenvalue of a graph is called main if its eigenspace is not orthogonal to the all-ones vector. Introduced by Cvetkovi\'{c} in the early 1970s and systematically studied by Rowlinson and others, graphs with exactly one or two main eigenvalues are now well understood. However, the classification of graphs with preci...
For a connected regular graph G and a vertex a, we study the algebra generated by the adjacency and degree matrices of G-a and its cyclic module P_a generated by the all-ones vector. Our main theorem determines dim P_a for Cartesian products whose factors have equitable distance partitions at the chosen roots. A normal...
In this paper, we study the degree of irrationality of properly elliptic surfaces with a section. We prove $\min\{\chi(\mathcal O_S),\,2\operatorname{gon}(C)\} \leq \operatorname{irr}(S) \leq 2\operatorname{gon}(C)$. The lower bound is obtained from the canonical bundle formula and the Cayley--Bacharach property. We sh...
