Moore Polygons – Spectral Extremals

• Moore Polygons are distance-regular graphs that achieve the spectral Moore bound, linking combinatorial structure with eigenvalue distribution.
• They are defined by a strict interplay of regularity, spectral gap, and diameter, exemplifying strong spectral expansion below the Ramanujan threshold.
• Their analysis employs linear programming methods to construct optimal polynomial bounds and utilizes nonexistence results to limit higher diameter cases.

A Moore polygon is a distance-regular graph that attains the spectral Moore bound, which is a rigorous upper limit for the maximum order $v(k,\theta)$ of a connected $k$-regular graph whose second-largest adjacency eigenvalue $\lambda_2$ is at most $\theta$. Moore polygons synthesize combinatorial and spectral graph theory, representing the extremal case in spectral expansion and regularity of finite graphs. Their study is foundational for understanding the interplay between combinatorial structure, eigenvalue distribution, and extremal graph size (Cioabă et al., 10 Dec 2025).

1. Definition and Spectral Characterization

Fix integer $k \geq 3$ and real threshold $\theta$. Define $v(k,\theta)$ as the largest $|V(G)|$ for any connected $k$-regular graph $G$ with $\lambda_2(G) \leq \theta$. Classical results show $v(k,\theta)$ is finite when $\theta < 2\sqrt{k-1}$ (Alon–Boppana/Serre), and infinite otherwise (Marcus–Spielman–Srivastava 2015). A graph achieving $v(k,\theta)$ is necessarily distance-regular, with intersection array of the form $\{k, k-1, ..., k-1;\;1,1,...,1,c\}$.

The spectral Moore bound is given by:

$$v(k,\theta) \leq M(k,t,c) = 1 + \sum_{i=0}^{t-3} k(k-1)^i + \frac{k(k-1)^{t-2}}{c}$$

where $t \geq 3$ is an integer, $0 < c \leq k$, and $\theta$ is the second-largest eigenvalue of the tridiagonal matrix $T(k,t,c)$. Equality holds if and only if $G$ is a Moore polygon of diameter $t-1$.

2. Spectral Bounds and Infinite Families

Two key theorems delineate the transition between finite and infinite families:

• Alon–Boppana–Serre Theorem: For an infinite family of connected $k$-regular graphs $G_n$, $$\liminf_{n\to\infty} \lambda_2(G_n) \geq 2\sqrt{k-1}$$; thus, $v(k,\theta)<\infty$ whenever $\theta<2\sqrt{k-1}$.
• Marcus–Spielman–Srivastava (MSS): Exist infinite families with $\lambda_2$ arbitrarily close to $2\sqrt{k-1}$; for $\theta\geq2\sqrt{k-1}$, $v(k,\theta)=\infty$ holds by eigenvalue interlacing.

This dichotomy establishes Moore polygons as finite, discrete objects associated with strong spectral expansion below the Ramanujan threshold.

3. Linear-Programming Bound and Construction

Nozaki’s linear-programming (LP) method constructs a real polynomial $f(x)$ in terms of orthogonal polynomials $F_i^k(x)$ defined via the non-backtracking walk relation:

• $F_0^k(x) = 1$
• $F_1^k(x) = x$
• $F_i^k(x) = xF_{i-1}^k(x) - (k-1)F_{i-2}^k(x)$ for $i\geq2$.

Given certain positivity and monotonicity constraints on $f(x)$ over the eigenvalues, one shows $|V(G)| \leq f(k)/f_0$. Optimal choices, reflecting the eigenstructure of the quotient matrix $T(k,t,c)$ from distance partitioning, yield the spectral Moore bound, which tightens classical Moore bounds and underlies the definition of Moore polygons.

4. Nonexistence Results for Moore Polygons

There are extensive nonexistence results:

• Diameter $\geq 6$: Damerell–Georgiacodis proved no Moore polygons exist for diameter $d\geq6$ ($t\geq7$).
• Diameter 3, $\theta=\sqrt{k}$: For intersection array $\{k, k-1, k-1; 1, 1, k-\sqrt{k}\}$, the required eigenvalue equation forces non-integer roots unless $k=4$, with the only case being the Odd graph $O_4$.
• Diameter 4, $\theta=\sqrt{2k-1}$: The array $\{k, k-1, k-1, k-1; 1, 1, 1, k-\sqrt{2k-1}\}$ leads to a cubic polynomial whose roots cannot all be integer for $k \geq 3$.

The cumulative effect is that Moore polygons with second-largest eigenvalue $\sqrt{k}$ or $\sqrt{2k-1}$ do not exist for $k\geq3$ except for known special cases.

5. Exact Values and Concrete Constructions

Explicit extremal values for $v(k,\theta)$ and corresponding Moore polygons include:

Parameters	Value	Extremal Graph
$v(4, \sqrt{2})$	$14$	co-Heawood (bipartite, 4-regular)
$v(5, \sqrt{5}-1)$	$16$	folded $5$-cube
$v(5, \sqrt{2})$	$16$	folded $5$-cube

• The co-Heawood graph is a bipartite, 4-regular, distance-regular graph with spectrum $\{4^1, \sqrt{2}^6, -\sqrt{2}^6, -4^1\}$.
• The folded $5$-cube is strongly regular with parameters $(16,5,0,2)$ and $\lambda_2=1$.

For $k=5$, a spectral gap occurs: any connected $5$-regular graph with $\lambda_2>1$ must satisfy $\lambda_2\geq\sqrt{5}-1$, with the unique extremal $10$-vertex Cayley graph $\mathrm{Cay}(\mathbb{Z}_{10},\{\pm1,\pm2,5\})$ attaining equality.

6. Uniqueness, Spectral Jumps, and Strengthened Bounds

• Spectral Uniqueness: For $\lambda_2=\sqrt{5}-1$ and $k=5$, only one connected graph on $10$ vertices attains this—$\mathrm{Cay}(\mathbb{Z}_{10},\{\pm1,\pm2,5\})$.
• Spectral Jump: For $k=5$, graphs with $\lambda_2(G)>1$ experience a jump to at least $\sqrt{5}-1$. This demarcates a strong spectral barrier within the family of regular graphs.
• Strengthened Alon–Boppana (Kolokolnikov’s conjecture): For connected $k$-regular graphs of order $n\geq\frac{2(k-1)^d-2}{k-2}$, one obtains $$\lambda_2(G)\geq2\sqrt{k-1}\cos\left(\frac{\pi}{d}\right)$$, extending and refining previous spectral diameter bounds, especially for cubic graphs.

7. Implications and Connections in Extremal Graph Theory

Moore polygons coincide precisely with distance-regular graphs with intersection arrays as above, saturating the spectral Moore bound. Their nonexistence results contribute to the longstanding classification challenge for small-diameter distance-regular graphs. The extremal values $v(k,\theta)$ provide connections between Ramanujan expansion theory (MSS, Alon–Boppana) and classical combinatorial bounds (degree/distance/diameter/Moore).

The LP approach generalizes to bipartite graphs and hypergraphs, generating new algebraic connectivity bounds and girth-based spectral lower bounds. The spectral Moore problem thus unifies research threads in algebraic combinatorics: distance-regularity, eigenvalue interlacing, LP techniques, and extremal spectral graph theory. These results clarify the boundary between finite and infinite families of near-Ramanujan graphs and deepen understanding of structural limits imposed by spectral expansion (Cioabă et al., 10 Dec 2025).