[Paper Review] Non-singular circulant graphs and digraphs

A. K. Lal, A. Satyanarayana Reddy|arXiv (Cornell University)|Jun 4, 2011

Graph theory and applications4 references3 citations

TL;DR

This paper provides necessary and sufficient conditions for the singularity of generalized circulant graphs and digraphs, particularly $(r,s,t)$-digraphs and $C_n^{i,j,k,l}$ digraphs, using properties of circulant matrices, cyclotomic polynomials, and roots of unity. The key contribution is a precise characterization of singularity based on gcd conditions involving $n$, $r$, $s$, $t$, $j$, $k$, and $l$, extending known results on circulant matrix eigenvalues and determinants.

ABSTRACT

We give necessary and sufficient conditions for a few classes of known circulant graphs and/or digraphs to be singular. The above graph classes are generalized to $(r,s,t)$-digraphs for non-negative integers $r,s$ and $t$, and the digraph $C_n^{i,j,k,l}$, with certain restrictions. We also obtain a necessary and sufficient condition for the digraphs $C_n^{i,j,k,l}$ to be singular. Some necessary conditions are given under which the $(r,s,t)$-digraphs are singular.

Motivation & Objective

• To establish necessary and sufficient conditions for the singularity of known classes of circulant graphs and digraphs.
• To generalize existing circulant graph families into $(r,s,t)$-digraphs and $C_n^{i,j,k,l}$ digraphs with defined structural constraints.
• To derive a precise characterization of singularity for $C_n^{i,j,k,l}$ digraphs using algebraic number theory and matrix theory.
• To provide necessary conditions under which $(r,s,t)$-digraphs are singular, advancing beyond partial results.
• To unify and extend prior results on circulant matrix determinants and eigenvalues via representer polynomials and roots of unity.

Proposed method

• Utilizes the representer polynomial $\gamma_A(x)$ of circulant matrices to express adjacency matrices of generalized circulant graphs.
• Applies Lemma 1.4 to link matrix singularity to the degree of $\gcd(\gamma_A(x), x^n - 1)$, which must be at least 1 for singularity.
• Employs cyclotomic polynomials $\Phi_d(x)$ and properties of primitive roots of unity $\zeta_n$ to analyze eigenvalues and nullity.
• Derives the representer polynomial for $C_n^{i,j,k,l}$ as $x^i \cdot \frac{x^{\ell+1}-1}{x-1} \cdot \frac{x^{(k+1)j}-1}{x^j-1}$, enabling algebraic analysis.
• Uses algebraic number theory to test whether $\gamma_A(\zeta_n^{n/d}) = 0$ for some $d \geq 2$, indicating a zero eigenvalue and thus singularity.
• Applies the identity $(x-1)\gamma_A(x) = (x^r - 1) + x^{r+t}(x^s - 1)$ to analyze roots and derive singularity conditions for $(r,s,t)$-digraphs.

Experimental results

Research questions

• RQ1 Under what conditions is a generalized $(r,s,t)$-digraph singular?
• RQ2 When is the circulant digraph $C_n^{i,j,k,l}$ singular, given constraints on $i,j,k,l,n$?
• RQ3 How do gcd conditions on $n$, $r$, $s$, $t$, $j$, $k$, and $\ell$ determine the singularity of these digraphs?
• RQ4 Can the singularity of $C_n^{i,j,k,l}$ be fully characterized using cyclotomic polynomial divisibility?
• RQ5 What necessary conditions ensure singularity in the broader class of $(r,s,t)$-digraphs?

Key findings

• The digraph $C_n^{i,j,k,l}$ is singular if and only if $\gcd(\ell+1, n) \geq 2$ or $\gcd(k+1, \frac{n}{\gcd(n,j)}) \geq 2$.
• For $(r,s,t)$-digraphs with first row $[r$ times $a$, $s$ times $b$, $t$ times $c]$, the matrix is singular if $\gcd(s,n) > 1$.
• If $\gcd(n,r,s) > 1$, then the $(r,s,t)$-digraph is singular, as $\zeta_n^{n/k}$ is a root of the representer polynomial.
• When $\gcd(n,s) = 1$ and $d \geq 2$ divides $t$ with $s = \ell r$ and $\ell \equiv -1 \pmod{d}$, the digraph is singular due to $\gamma_A(\zeta_n^{n/d}) = 0$.
• If $n = 2m$, $\gcd(n,s) = 1$, and $d$ is even with $r+t$ an odd multiple of $d/2$ and $s = \ell r$ with $\ell \equiv 1 \pmod{d}$, then $\gamma_A(\zeta_n^{n/d}) = 0$, implying singularity.
• The complete graph $K_n$ is non-singular for $n \geq 2$, as confirmed by Corollary 1.5 with $a=1$, $b=0$, $s=n-1$, $t=1$, and $\gcd(s,n)=1$.