Distance-Regular Graphs of Negative Type

• Distance-regular graphs of negative type are characterized by uniform intersection arrays and a distance matrix with exactly one positive eigenvalue, emphasizing their unique metric structure.
• Their spectral properties, such as a smallest eigenvalue at most -k/2 and half-antipodal behavior in the Kneser graph, provide a clear criterion for classification.
• These graphs connect deeply with finite geometry and association schemes, with classical families like dual polar and Hermitian forms illustrating their rigid structural constraints.

A distance-regular graph of negative type is a finite, connected graph whose combinatorial structure enforces uniform local regularities and, crucially, whose spectral or metric properties exhibit specific negative behaviors. This notion appears in several equivalent guises: as distance-regular graphs with smallest eigenvalue at most $-k/2$ (where $k$ is the valency), as those whose distance-$d$ graph (“Kneser graph”) has all negative eigenvalues equal (half-antipodal case), or as distance-regular graphs for which the distance matrix has inertia $(1,*,n-1-*)$—that is, exactly one positive eigenvalue. They are important in algebraic graph theory for their rigidity, connections to association schemes, and links to finite geometry and group theory.

1. Definitions and Characterizations

A connected graph $\Gamma$ with diameter $d$ is distance-regular if there exist constants $b_0,b_1,\ldots,b_{d-1},c_1=1,c_2,\ldots,c_d$ (the intersection array) such that for distinct vertices $x$, $y$ at distance $i$, the number of neighbors of $y$ at distance $i-1$, $i$, $i+1$ from $x$ is $c_i$, $a_i = b_0-b_i-c_i$, $b_i$, respectively. Equivalently, all distance-$i$ graphs $\Gamma_i$ have adjacency matrices that are polynomials in $A_1$ of degree exactly $i$, encoded by predistance polynomials $p_i$.

Negative type arises in three intertwined senses:

• Spectral: The smallest adjacency eigenvalue $\theta_{\min}$ satisfies $\theta_{\min} \leq -k/2$; the graph is then called of negative (or half-negative) type (Koolen et al., 2015).
• Distance spectrum: The distance matrix $D$ has exactly one positive eigenvalue, with all others non-positive. This is the negative type property in metric geometry (Aalipour et al., 2015).
• Kneser graph eigenvalues: The distance-$d$ graph $\Gamma_d$ has all negative eigenvalues equal (the half-antipodal/negative-type phenomenon), captured precisely by spectra and predistance polynomials (Fiol, 2014).

2. Classification and Core Families

The structure of distance-regular graphs of negative type is tightly constrained. For small diameter $D\leq 4$, all such graphs can be classified explicitly, with only finitely many possible valencies for any fixed diameter $D$ (Koolen et al., 2015). In the classical setting (parameterized by $(d, b, \alpha, \beta)$), for $b < -1$ and $d \geq 4$, only two infinite families persist:

Family	Parameters	Negative type condition
Dual polar $Q^-(2d-1,q^2)$	$(d, -q, -q(q+1)/(q-1), -q((−q)^d+1)/(q-1))$	$b = -q < -1$
Hermitian forms $H(d,q^2)$	$(d, -q, -q-1, -((-q)^d+1))$	$b = -q < -1$

No further classical families of negative type exist for $d\geq4$; a hypothetical third family has been ruled out by reduction to $d=3$ and recent nonexistence results (Adriaensen et al., 19 Nov 2025). For small diameters, sporadic examples exist, such as the Petersen, Sylvester, and Hoffman–Singleton graphs.

3. Spectral Theory and the Spectral Excess Theorem

Distance-regular graphs of negative type possess rigid spectral regularities. The eigenvalues $\lambda_0 = k > \lambda_1 > \cdots > \lambda_d$ are precisely determined by the intersection array. The distance-$d$ graph—Kneser graph $K = \Gamma_d$—is governed by the predistance polynomial $p_d(A)$. The half-antipodal condition asserts that all odd-index $p_d(\lambda_i)$ are equal to a negative value $\tau$, creating a spectral collapse on the negative side.

The spectral excess theorem gives a powerful criterion: for any regular graph, the average number $\overline{k_d}$ of vertices at maximal distance can be bounded in terms of the spectrum and the combinatorial data. In the half-antipodal/negative-type case, equality is achieved precisely by distance-regular graphs with collapsed negative eigenvalues, leading to explicit spectral characterizations of negative type (Fiol, 2014).

4. Metric Perspective: Distance Matrices and Inertia

From the metric viewpoint, a distance-regular graph is of negative type if its distance matrix $D$ has inertia $(1, n_0, n-1-n_0)$ (precisely one positive eigenvalue). Koolen–Shpectorov completed the classification of such graphs among distance-regular graphs:

• Cocktail-party graphs $CP(m)=K_{2,\dots,2}$
• Cycles $C_n$
• Hamming graphs $H(d,n)$
• Doob graphs $D(m,4)$
• Johnson graphs $J(n,r)$
• Double odd graphs $DO(r)$
• Halved cubes $\tfrac12Q_d$
• Six sporadic graphs: Gosset, Schlӓfli, three Chang, icosahedral, Petersen, and dodecahedral (Aalipour et al., 2015)

Each family has explicit distance spectra, with unique positive roots and all other eigenvalues non-positive.

5. Connections to Finite Geometry and Association Schemes

Many distance-regular graphs of negative type are realized as geometric objects or are closely linked to finite classical groups. Dual polar graphs arise from Hermitian polar spaces, and the Hamming and Johnson graphs arise from combinatorial geometries and association schemes. Q-polynomial property holds for all negative-type graphs, reflecting deep regularity in the underlying association schemes (Koolen et al., 2015).

In the classical setting, the only negative-type graphs with $d\ge4$ and $a_1\neq0$, $c_2>1$ are $Q^-(2d-1,q^2)$ and $H(d,q^2)$; no graphs of the so-called “last” parameter family exist. This linkage constrains the existence of certain configurations in finite geometry: for example, the nonexistence of hemisystems in $H(2d-1,q^2)$ for odd $q>3$ is a corollary (Adriaensen et al., 19 Nov 2025).

6. Examples, Extremal Graphs, and Open Problems

Notable infinite families and sporadic instances include odd cycles, complete tripartite graphs $K_{t,t,t}$, folded odd cubes, the odd graphs $O_{D+1}$, and the aforementioned Hamming, Johnson, and dual polar graphs. Each is characterized by explicit intersection arrays and spectral data.

Outstanding problems include full classification for $a_1=1$ (“geometric” graphs) in all diameters, the D=4 non-geometric negative-type cases, and extension of classification below the $-k/2$ threshold. Whether new infinite families exist outside those currently known remains open, but current results imply strong finiteness for fixed diameter and threshold.

7. Proof Techniques and Theoretical Tools

Classification and structural results rely on:

• Spectral techniques: Biggs’ multiplicity formula, eigenvalue interlacing, Krein parameters and absolute bounds (Koolen et al., 2015, Fiol, 2014).
• Predistance polynomial analysis: Coalescence of $p_d(\lambda_i)$ in half-antipodal regimes.
• Association scheme diagonalization: For families such as Johnson and Hamming graphs (Aalipour et al., 2015).
• Feasibility and integrality checks: Ensuring intersection numbers and Krein parameters are valid for candidate arrays.
• Geometric and combinatorial reductions: Use of induced subgraphs to eliminate hypothetical families.
• Link to finite geometry: Transfer of graph-theoretic results to the existence (or nonexistence) of structures like hemisystems.

These elements jointly yield a highly constrained landscape for distance-regular graphs of negative type, integrating spectral, metric, and geometric perspectives with group- and scheme-theoretic methods.