[Paper Review] The spectra of generalized Paley graphs of $(q^\ell+1)$-th powers and applications

Ricardo A. Podestá, Denis E. Videla|arXiv (Cornell University)|Dec 8, 2018

Finite Group Theory Research31 references4 citations

TL;DR

This paper introduces a new family of generalized Paley graphs based on $(q^{ u}+1)$-th powers in finite fields $\mathbb{F}_{q^m}$, computes their exact spectra and energy, and proves they are integral, strongly regular, and non-bipartite. The key contribution is constructing infinite families of integral Ramanujan graphs in all characteristics and solving Waring’s problem for exponents $q^\ell+1$, showing $g(q^\ell+1, q^m) = 1$ or $2$. The results also yield equienergetic non-isospectral graphs and explicit Ihara zeta functions.

ABSTRACT

We consider a special class of generalized Paley graphs over finite fields, namely the Cayley graphs with vertex set $\mathbb{F}_{q^m}$ and connection set the nonzero $(q^\ell+1)$-th powers in $\mathbb{F}_{q^m}$, as well as their complements. We explicitly compute the spectrum and the energy of these graphs. As a consequence, the graphs turn out to be (with trivial exceptions) simple, connected, non-bipartite, integral and strongly regular, of pseudo or negative Latin square type. By using the spectral information we compute several invariants of these graphs. We exhibit infinitely many pairs of equienergetic non-isospectral graphs. As applications, on the one hand we solve Waring's problem over $\mathbb{F}_q^m$ for the exponents $q^\ell+1$, for each $q$ and for infinitely many values of $\ell$ and $m$. We obtain that the Waring's number $g(q^\ell+1,q^m)=1$ or $2$, depending on $m$ and $\ell$, thus solving some open cases. On the other hand, we construct infinite towers of Ramanujan graphs in all characteristics. Finally, we give the Ihara zeta functions of these graphs.

Motivation & Objective

• To study spectral and structural properties of a new class of generalized Paley graphs defined by $(q^\ell+1)$-th powers in $\mathbb{F}_{q^m}$.
• To construct infinite families of integral, non-bipartite, Ramanujan graphs in all characteristics.
• To solve Waring’s problem over $\mathbb{F}_{q^m}$ for exponents $q^\ell+1$, determining the Waring number $g(q^\ell+1, q^m)$.
• To produce infinite pairs of integral equienergetic non-isospectral graphs.
• To compute the Ihara zeta functions of these graphs and derive their complexity via zeta function identities.

Proposed method

• Define Cayley graphs $\Gamma_{q,m}(\ell)$ with vertex set $\mathbb{F}_{q^m}$ and connection set the nonzero $(q^\ell+1)$-th powers in $\mathbb{F}_{q^m}^*$.
• Use exponential sums over quadratic forms $Q_{\gamma,\ell}(x) = \operatorname{Tr}_{q^m/q}(\gamma x^{q^\ell+1})$ to compute eigenvalues and multiplicities.
• Apply known results on ranks and types of quadratic forms over finite fields to classify the spectrum of $\Gamma_{q,m}(\ell)$.
• Derive the spectrum of the complement graphs $\bar{\Gamma}_{q,m}(\ell)$ using spectral duality and eigenvalue complementarity.
• Use the determinant formula for the Ihara zeta function $\zeta_\Gamma(t)^{-1} = (1-t^2)^{e-n} \prod_{i=1}^n Q(\lambda_i, t)$ with $Q(s,t) = 1 - st + (k-1)t^2$.
• Compute the complexity $K(\Gamma)$ via Kirchhoff’s formula using the zeta function and eigenvalue data.

Experimental results

Research questions

• RQ1 What is the spectrum of the Cayley graph $\Gamma_{q,m}(\ell)$ defined by $(q^\ell+1)$-th powers in $\mathbb{F}_{q^m}$?
• RQ2 For which parameters is $\Gamma_{q,m}(\ell)$ a Ramanujan graph, and can such graphs be constructed in all characteristics?
• RQ3 What is the Waring’s number $g(q^\ell+1, q^m)$, and does it equal 1 or 2 for infinitely many $q$, $\ell$, $m$?
• RQ4 Can one construct infinite families of integral, equienergetic, non-isospectral graphs using this construction?
• RQ5 What is the explicit form of the Ihara zeta function and complexity of $\Gamma_{q,m}(\ell)$ and its complement?

Key findings

• The graphs $\Gamma_{q,m}(\ell)$ are integral, strongly regular, and non-bipartite (except for trivial cases), with exactly 2 or 3 distinct eigenvalues depending on whether $\ell = m/2$.
• For $\ell \neq m/2$, the spectrum is $\{[k]^1, [\upsilon]^{m_\upsilon}, [\mu]^{m_\mu}\}$ with integer eigenvalues $k$, $\upsilon$, $\mu$, and multiplicities $m_\upsilon$, $m_\mu$.
• The Waring’s number $g(q^\ell+1, q^m)$ is 1 if $m$ is odd and 2 if $m$ is even, solving open cases for infinitely many $q$, $\ell$, $m$.
• The paper constructs infinite families of integral Ramanujan graphs in all characteristics, including non-bipartite ones.
• Explicit Ihara zeta functions are derived, with $\zeta_\Gamma(t)^{-1}$ expressed as a product over eigenvalues using $Q(s,t) = 1 - st + (k-1)t^2$.
• Complexity $K(\Gamma)$ is computed as $\frac{1}{q^m}(k - \upsilon)^{m_\upsilon}(k - \mu)^{m_\mu}$, and verified via zeta function limits.