[Paper Review] Distance spectra and Distance energy of Integral Circulant Graphs

Aleksandar Ili ' c|arXiv (Cornell University)|Apr 6, 2011

Graph theory and applications22 references3 citations

TL;DR

This paper characterizes the distance spectra and distance energy of integral circulant graphs (ICGs), proving that their distance matrices have integral eigenvalues. It derives exact formulas for the distance energy of unitary Cayley graphs and constructs two infinite families of non-cospectral, distance equienergetic ICGs—one where one graph is a subgraph of the other, and another with diameter three—using number-theoretic properties of divisors and Ramanujan sums.

ABSTRACT

The distance energy of a graph $G$ is a recently developed energy-type invariant, defined as the sum of absolute values of the eigenvalues of the distance matrix of $G$. There was a vast research for the pairs and families of non-cospectral graphs having equal distance energy, and most of these constructions were based on the join of graphs. A graph is called circulant if it is Cayley graph on the circulant group, i.e. its adjacency matrix is circulant. A graph is called integral if all eigenvalues of its adjacency matrix are integers. Integral circulant graphs play an important role in modeling quantum spin networks supporting the perfect state transfer. In this paper, we characterize the distance spectra of integral circulant graphs and prove that these graphs have integral eigenvalues of distance matrix $D$. Furthermore, we calculate the distance spectra and distance energy of unitary Cayley graphs. In conclusion, we present two families of pairs $(G_1, G_2)$ of integral circulant graphs with equal distance energy -- in the first family $G_1$ is subgraph of $G_2$, while in the second family the diameter of both graphs is three.

Motivation & Objective

• To characterize the distance spectra of integral circulant graphs (ICGs), proving their distance eigenvalues are integers.
• To compute the distance energy of unitary Cayley graphs using number-theoretic methods.
• To construct infinite families of non-cospectral, distance equienergetic ICGs without relying on graph products or line graphs.
• To explore structural and spectral properties of ICGs relevant to quantum spin networks and perfect state transfer.
• To extend existing results on equienergetic graphs by introducing novel constructions based on divisor sets and multiplicative functions.

Proposed method

• Utilizes the definition of ICG_n(D) as Cayley graphs over Z_n with edges defined by gcd(a−b,n) ∈ D, where D is a set of proper divisors of n.
• Applies Ramanujan sums c(r,n) = ∑_{d|gcd(r,n)} μ(d) · e^{2πi r/d} to compute distance eigenvalues μ_r via additive combinations.
• Employs the formula μ_r = ∑_{d∈D} c(r, n/d) to express distance eigenvalues in terms of Ramanujan sums and divisor structure.
• Uses the multiplicative property of Euler's totient function φ(n) and the structure n = 2^k·m to analyze cases and derive closed-form expressions.
• Applies spectral decomposition and multiplicity counting based on gcd(r,n) to compute distance energy as sum of absolute eigenvalues.
• Derives distance energy formulas for specific ICGs such as ICG_n(1) and ICG_n(1,p) by case analysis on r modulo divisors of n.

Experimental results

Research questions

• RQ1 What is the structure of the distance spectrum of integral circulant graphs, and are their distance eigenvalues always integers?
• RQ2 Can closed-form expressions for the distance energy of unitary Cayley graphs be derived using number-theoretic tools?
• RQ3 Are there infinite families of non-cospectral integral circulant graphs with equal distance energy that do not rely on graph products or line graphs?
• RQ4 How does the diameter of ICGs affect their distance energy, particularly in cases with diameter three?
• RQ5 Under what number-theoretic conditions do ICG_n(1) and ICG_n(1,p) have equal distance energy, and what does this imply for quantum network design?

Key findings

• The distance matrix of any integral circulant graph has integral eigenvalues, confirming that ICGs are D-integral.
• For unitary Cayley graphs ICG_n(1), the distance energy is DE(ICG_n(1)) = 6(φ(n)−1)·2 − n + 2(2−p)(2−q) when n=pq, simplifying to 6(p−1)(q−1) for odd primes p,q.
• A family of equienergetic ICGs exists where ICG_{3p}(1) is a subgraph of ICG_{3p}(1,p), and both have equal distance energy 6(p−1)(2) for p>3.
• For n=2pq with distinct odd primes p,q, the graphs ICG_{2pq}(1,p) and ICG_{2pq}(1,q) are non-cospectral and equienergetic with DE = 12pq − 4p − 4q − 4.
• The diameter of ICG_{2pq}(1,p) and ICG_{2pq}(1,q) is three, marking the first such equienergetic families not based on graph products or iterated line graphs.
• The distance eigenvalues are computed via Ramanujan sums and grouped by gcd(r,n), enabling exact multiplicity counting and energy summation.