Spectrally Minimal Graphs

Updated 2 February 2026

Spectrally Minimal Graphs are defined as graphs that achieve the smallest adjacency spectral radius among connected graphs with fixed order and diameter.
Techniques such as eigenvalue interlacing, Perron–Frobenius monotonicity, and subgraph transformations classify minimizers like quipus, banana graphs, and path structures.
Recent work by Lan and Shi fully characterizes minimizers for diameter 2(n-1)/3, revealing structural phase transitions as graph diameter increases.

A spectrally minimal graph is a graph that minimizes a specified spectral parameter, typically the adjacency matrix spectral radius, within a prescribed class such as connected graphs of order $n$ and diameter $d$ . This extremal concept, motivated by the classical Brualdi–Solheid problem, is central in spectral graph theory and exists in variants for different matrix spectra (adjacency, distance, Laplacian), but the focus here is on the minimal adjacency spectral radius relative to structural constraints like graph order and diameter. The precise characterization of minimizers, sometimes termed "minimizer graphs," often leads to the identification of special structured families such as quipus, banana graphs, stars, paths, or block-clique trees, depending on the parameter regime. The recent work on graphs of order $n$ and diameter $2(n-1)/3$, notably by Lan and Shi (Lan et al., 2014), fills one of the key unresolved gaps in the diameter–radius extremal landscape.

1. Formal Definition and Spectral Radius

Given a simple undirected graph $G$ on $n$ vertices and adjacency matrix $A(G)$ , the spectral radius $\rho(G)$ is the largest eigenvalue of $A(G)$ , equivalently the largest real root of its characteristic polynomial $\phi_G(\lambda) = \det(\lambda I - A)$ . For fixed $n \geq 1$ and $1 \leq d \leq n-1$ , the spectrally minimal graph for diameter $d$ is a connected graph $G$ on $n$ vertices with $\operatorname{diam}(G) = d$ such that

$\rho(G) = \min \{ \rho(H) : H\text{ connected},\ |V(H)|=n,\ \operatorname{diam}(H)=d \}.$

2. Known Minimizers in Specific Diameter Regimes

The systematic study of spectral-minimizing graphs with fixed diameter originated with van Dam and Kooij (2007) and subsequent works (Lan et al., 2014). For extremal diameter cases and certain fractional-diameter regimes, minimizers are precisely classified:

Diameter $1$: $K_n$ is uniquely minimal, $\rho(K_n) = n-1$ .
Diameter $2$: $S_n = K_{1,n-1}$ is uniquely minimal for large $n$ , $\rho(S_n) = \sqrt{n-1}$ .
Diameter close to $n$ ( $d = n-k$ , $k=1,\dots,8$ ): Minimized by trees of max degree 3, specifically open quipus or “banana” graphs. E.g., $d=n-1$ , the path $P_n$ yields $\rho(P_n)=2\cos(\pi/(n+1))$ .
Fractional diameters $d \in [n/2,2n/3-1]$ : Minimizers are closed quipus $C(n-d-1,n-d-1)$ , with explicit but transcendental formulas for $\rho$ .
Critical diameters $d = (2n-4)/3, (2n-3)/3$ : Mixture of open and closed quipus, with explicit parameterizations and shared spectral radii (see (Lan et al., 2014)).

These classifications are nontrivial; proof methods range from eigenvalue interlacing, Perron–Frobenius monotonicity, structural reductions to path-like trees, and local graph modifications (subdivision, branch-balancing).

3. Minimizer Graphs for $d=2(n-1)/3$

For $n=3k+1$ , with $d=2k = 2(n-1)/3$ , Lan & Shi (Lan et al., 2014) completely determine the minimizers as:

Family: Open quipus $P(i,i+j-1,j)$ for $(i,j)$ nonnegative with $i+j=k$ .
Structure: Each consists of two branch-vertices $u,v$ of degree 3, joined by an internal path of length $i+j-1$ ; at $u$ hangs a pendent path of length $i$ , at $v$ a second of length $j$ .
Vertex count: $3k+1$.
Diameter verification: The maximal distance is attained between endpoints of the two pendent paths, yielding $i + (i + j - 1) + j = 2k = d$ .

All graphs in this family share the same spectral radius $\rho_k = \rho(P(i,i+j-1,j))$ , and no more elementary closed-form is available. The value is characterized as the largest root of certain rational or recurrence equations that build control on tree eigenvalues by leveraging Hoffman–Smith subdivision lemmas and rooted-graph polynomial recurrences.

4. Structural Description and Key Techniques

The proof that only these open quipus are minimizers proceeds via three steps:

Spectral Typology: Woo–Neumaier lemma restricts minimizers with $2<\rho<3/\sqrt{2}$ to daggers, open quipus, or closed quipus. Dagger graphs and closed quipus are ruled out on diameter and spectral grounds.
Combinatorial Constraints: Rigorous linear inequalities on lengths of pendent and internal paths limit quipu parameters; only open quipus with two branch vertices and specific attachment pattern persist.
Spectral Comparison: For the remaining candidates, subdivision and rooted-graph t-functions (as in Lemma 2.10 (Lan et al., 2014)) strictly increase spectral radius except for the main family, confirming their minimality.

A central motif is the phase transition in minimizer structure as $d$ crosses $(2n-3)/3$ : closed quipus dominate prior to this threshold, open quipus appear for $d=2(n-1)/3$ .

5. Implications and Open Directions

This classification addresses the last previously unresolved “fractional” diameter regime, establishing a complete and explicit family of spectrally minimal graphs for $d=2(n-1)/3$ . The transition sequence—closed quipus for $d=(2n-4)/3$ , mixed for $d=(2n-3)/3$ , open quipus for $d=2(n-1)/3$ —suggests multiple structural phase changes as $d$ increases with $n$ . A plausible implication is that for the remaining intervals of $d$ , further families of balanced quipu-like graphs or trees of bounded degree may constitute minimizers, but rigorous uniform theory is lacking.

Open problems include:

Classifying minimizers for general fractional diameters outside the settled regimes, potentially via refined quipu constructions.
Obtaining closed-form or asymptotic expansions for $\rho_k$ as $k \to \infty$ , possibly leveraging continued-fraction theory entailed in the rational characteristic polynomial recurrences.
Exploring extremal spectra under related matrix functions, such as Laplacian or signless Laplacian, for prescribed diameter.

6. Broader Context: Spectral Minimizers and Parametric Families

The notion of spectrally minimal graphs extends to various combinatorial constraints (independence number (Choi et al., 2023), domination number (Liu et al., 2022), matching number (Liu et al., 26 Jan 2026), connectivity/cut structures (Zhang et al., 2010, Zhang et al., 20 Jan 2025)). For each, an explicit (often parametric, tree-based or block-based) family is identified as minimizers, characterized by high symmetry and balanced attachment of pendant paths or cycle lengths. The underlying principle in all cases is that balanced structures with controlled branch lengths and maximal path distance enforce a smallest spectral radius, a property formalized in the relevant Hoffman-type or interlacing lemmas.

This programme connects combinatorial structure to global spectral behavior and proves that minimality is achieved not by local sparsification or cycle removal alone, but via specific canonical configurations (quipus, stars, block-clique trees) tailored to the constraint.

References: The structural and spectral characterization herein is drawn from Lan & Shi (Lan et al., 2014), with supporting context from contemporary works on spectral extremality in graph families.