The algebraic degree of spectra of circulant graphs

Abstract

We investigate the algebraic degree of circulant graphs, i.e. the dimension of the splitting field of the characteristic polynomial of the associated adjacency matrix over the rationals. Studying the algebraic degree of graphs seems more natural than characterizing graphs with integral spectra only. We prove that the algebraic degree of circulant graphs on n vertices is bounded above by $\phi (n)/2$, where φ denotes Euler's totient function, and that the family of cycle graphs provides a family of maximum algebraic degree within the family of all circulant graphs. Moreover, we precisely determine the algebraic degree of circulant graphs on a prime number of vertices.

Introduction

Which graphs have integral spectra? This question was raised by Harary & Schwenk in their article from 1973/74 [2]. In the end of their article, they already remarked that this problem appears intractable. We consider the spectrum of a graph as the multiset of eigenvalues of its associated adjacency matrix. The quest of characterizing adjacency matrices for which all eigenvalues are integers seems to be a challenging project, there is no satisfying answer so far. Thus, it is common to restrict to special families of graphs and to determine all graphs with integral spectra within this family.

Since eigenvalues of graphs are algebraic integers, from a number-theoretical point of view, it seems more natural to ask the more general question: Which graphs have the same algebraic degree? This question was raised by Steuding, Stumpf and the author [6]. Given a graph G, the algebraic degree $\mathrm{deg}(G)$ is defined as the dimension of the splitting field of the characteristic polynomial of the associated adjacency matrix over the rationals. By definition, this splitting field is the smallest field that contains all eigenvalues of the spectrum of the graph.

Within the family of circulant graphs, So [8] completely characterized those with integral spectra. A graph is said to be circulant if it has a circulant adjacency matrix. Since each row is a cyclic shift of the first row, such a matrix is completely determined by specifying its first row. Therefore, with every circulant graph, we can associate a set $S\subseteq {\mathbb{Z}}_n$ (where ${\mathbb{Z}}_n$ denotes the ring of integers modulo n) of the positions of non-zero entries of the first row of the adjacency matrix of the graph. Respectively, we denote by ${〈S〉}_n$ the corresponding graph and call S the connection set of ${〈S〉}_n$. These notations are adopted from Mans et al. [4], [5]. Two vertices $x,y\in {\mathbb{Z}}_n$ are adjacent in ${〈S〉}_n$ if and only if $x-y\in S$. All graphs in this paper are assumed to be simple, i.e. undirected and without loops or multi-edges. Note that a circulant graph ${〈S〉}_n$ is undirected if and only if S is symmetric, i.e. $S\equiv -S\mathrm{mod}n$, and ${〈S〉}_n$ has no loops if and only if $0\notin S$. In view of this, we call a set $S\subseteq {\mathbb{Z}}_n$ a connection set if $S\equiv -S\mathrm{mod}n$ and $0\notin S$. The complete graph $K_n\cong {〈\{1,\dots ,n-1\}〉}_n$ and the cycle graph $C_n\cong {〈\{1,n-1\}〉}_n$ are well-known examples of circulant graphs. The spectrum of a circulant graph ${〈S〉}_n$ is given by

$$\mathrm{spec}({〈S〉}_n)=\{\sum _{s\in S}\mathrm{e}{\left(\frac{s}{n}\right)}^k|0\le k\le n-1\},$$

where here and in the following $\mathrm{e}\left(x\right)$ denotes $\exp (2\pi ix)$. A proof for this can be found in the book by Zhang [9]. In particular, circulant graphs are Cayley graphs on cyclic groups, i.e. ${〈S〉}_n\cong \mathrm{Cay}({\mathbb{Z}}_n,S)$.

In this paper, we precisely determine the algebraic degree of ${〈S〉}_p$, where $S\subseteq {\mathbb{Z}}_p$ is a connection set and p is a prime number. We show that the algebraic degree of ${〈S〉}_p$ is bounded from below by a function depending on the number of elements in S only. We also give bounds for the algebraic degree of circulant graphs on n vertices where n is any natural number. Furthermore, we show that the family of cycle graphs provides a family of maximum algebraic degree within the family of all circulant graphs, and give a lower bound for the number of non-isomorphic circulant graphs on a prime number of vertices and of maximum algebraic degree. Graphs of maximum algebraic degree can be considered as a counterpart of graphs with integral spectra.

Throughout this paper, by φ we denote Euler's totient function and ${\mathbb{Z}}_n^{⁎}$ denotes the group of units in ${\mathbb{Z}}_n$.