---
title: Amply Regular Graphs
url: https://www.emergentmind.com/topics/amply-regular-graphs
type: topic
---

# Amply Regular Graphs

Searching arXiv for recent papers on amply regular graphs to ground the article.
An amply regular graph is a connected regular graph in which local common-neighbor counts are prescribed at distances \(1\) and \(2\). In the standard notation \((v,k,\lambda,\mu)\), it is a \(k\)-regular graph on \(v\) vertices such that any two adjacent vertices have exactly \(\lambda\) common neighbors and any two vertices at distance \(2\) have exactly \(\mu\) common neighbors. This class properly contains strongly regular graphs, which are exactly the diameter-\(2\) case, and includes all non-complete distance-regular graphs through the identifications \(k=b_0\), \(\lambda=b_0-b_1-1\), and \(\mu=c_2\) [2210.02988], [2602.10396]. Recent work has made amply regular graphs a focal point for interactions among local combinatorics, discrete curvature, spectral theory, diameter bounds, edge-connectivity, clique structure, and constructions from designs, association schemes, and finite-field functions [2410.21055], [2507.18120], [1809.03422].

## 1. Definition, notation, and basic position in graph theory

A graph \(\Gamma\) on \(v\) vertices is called amply regular with parameters \((v,k,\lambda,\mu)\) if \(\Gamma\) is \(k\)-regular, any two adjacent vertices have exactly \(\lambda\) common neighbors, and any two vertices at distance \(2\) have exactly \(\mu\) common neighbors [2210.02988]. In the alternative notation \((n,d,\alpha,\beta)\), one writes \(v=n\), \(k=d\), \(\lambda=\alpha\), and \(\mu=\beta\) [2602.10396]. The class is local in nature: its defining conditions constrain only spheres of radius at most \(2\), and therefore amply regular graphs may have diameter greater than \(2\), finite or infinite order, and a wide range of global structures [2410.21055], [2507.18120].

Strongly regular graphs are precisely the amply regular graphs of diameter \(2\); for them, “non-adjacent” and “distance \(2\)” coincide [1809.03422]. Every non-complete distance-regular graph is amply regular, with parameter identification \((v,k,\lambda,\mu)=(n,b_0,b_0-b_1-1,c_2)\) [2602.10396]. Terwilliger graphs form a distinguished subclass: a non-complete graph is a Terwilliger graph if for any vertices \(x,y\) at distance \(2\), the common neighbors of \(x\) and \(y\) induce a clique of size \(\mu\); in amply regular graphs this is equivalent to having no induced quadrangles [2602.10396].

Standard local notation around an edge \(xy\) is central. Writing \(\Gamma(x)\) for the neighborhood of \(x\), one sets
\[
\Delta_{xy}:=\Gamma(x)\cap \Gamma(y),
\]
and
\[
N_x:=\Gamma(x)\setminus (\{y\}\cup \Gamma(y)),\qquad
N_y:=\Gamma(y)\setminus (\{x\}\cup \Gamma(x)).
\]
Then \(|\Delta_{xy}|=\lambda\) and \(|N_x|=|N_y|=k-\lambda-1\) [2210.02988]. These sets drive most modern arguments on curvature and matching.

A basic universal inequality relates the parameters:
\[
k(k-\lambda-1)\le (v-k-1)\mu,
\]
with equality if and only if the graph is strongly regular [2409.06418]. Additional structural inequalities appear when the diameter is at least \(3\) or \(4\). For amply regular graphs with diameter \(D\ge 4\), one has \(\mu\le k/2\), with equality if and only if the graph is either a polygon or a Hadamard graph; also \(b_1=k-\lambda-1\ge (k+1)/3\), hence \(\lambda\le (2/3)(k-2)\) [2605.25754]. Other bounds used in curvature-based analyses include
\[
d\ge 2\alpha-\beta+3,
\]
with equality iff the graph is the icosahedron or a line graph of a regular graph of girth at least \(5\), and, if \(D\ge 4\),
\[
d\ge 2\beta
\]
[2410.21055].

## 2. Local combinatorics, matching structure, and neighborhood geometry

Much of the modern theory proceeds by encoding the local combinatorics around an edge \(xy\) into auxiliary bipartite graphs whose perfect matchings control transport and curvature. In the girth-\(3\) case with \(\beta>\alpha\ge 1\), a transport-bipartite graph \(H=(V_H,E_H)\) can be built from \(N_x\), \(N_y\), the common-neighbor set \(\Delta_{xy}\), a copy \(\Delta'_{xy}\), and balancing vertices \(x_i,x'_i\) when \(\beta-\alpha>1\) [2210.02988]. Its edge set consists of eight explicitly defined families \(E_1,\dots,E_8\), and one checks that every vertex has degree \(\beta-1\); thus \(H\) is \((\beta-1)\)-regular and decomposes into \((\beta-1)\) edge-disjoint perfect matchings by Kőnig’s theorem [2210.02988].

A different but related perspective appears in the study of sharp curvature upper bounds. For an edge \(xy\) in a \(d\)-regular graph, the Lin–Lu–Yau curvature satisfies
\[
\kappa_{\mathrm{LLY}}(x,y)\le \frac{2+|\Delta_{xy}|}{d},
\]
and equality holds iff there exists a perfect matching between \(N_x\) and \(N_y\) [2410.21055]. For conference graphs and broader amply regular classes, local perfect matchings are forced by Hall-type arguments derived from counting common neighbors across the decomposition
\[
V=\{x\}\cup\{y\}\cup A_{xy}\cup N_x\cup N_y\cup P_{xy},
\]
where \(A_{xy}=\Gamma(x)\cap\Gamma(y)\) and \(P_{xy}=V\setminus (\Gamma(x)\cup \Gamma(y))\) [2409.06418]. The key observation is that common-neighbor counts yield quadratic inequalities in the total number of edges from a Hall-obstruction subset \(S\subseteq N_x\) into \(N_y\), and these inequalities rule out Hall failure under explicit parameter regimes [2409.06418].

This matching viewpoint also clarifies why local parameters alone do not always determine global geometry. In the strongly regular parameter set \((16,6,2,2)\), the Shrikhande graph and the \(4\times 4\) Rook’s graph have different curvature values, even though their \((n,k,\alpha,\beta)\) parameters coincide [2210.02988]. This suggests that local matching structure inside and around \(\Delta_{xy}\), rather than the parameter quadruple alone, can be decisive.

A further rigidity phenomenon arises in the regime \(\mu=(k-1)/2\) with diameter at least \(4\). Jin–Koolen–Lv prove that connected amply regular graphs with \(k\ge 5\) odd, \(d\ge 4\), and
\[
\mu=\frac{k-1}{2}
\]
must satisfy \(\lambda=0\) and \(d\le 5\), and are then exactly one of three types: the \(5\)-cube, the graph \(\mathbf{K}_2\square \Lambda\) where \(\Lambda\) is the unique bipartite \((0,2)\)-graph on \(14\) vertices, or the point-block incidence graph of a \(GDDDP(2,k+1;\,k;\,0,\frac{k-1}{2})\) [2605.25754]. In the last case the graph is bipartite, has diameter \(4\), and admits an equitable distance partition with quotient matrix
\[
Q=
\begin{pmatrix}
0 & k & 0 & 0 & 0\\
1 & 0 & k-1 & 0 & 0\\
0 & \frac{k-1}{2} & 0 & \frac{k+1}{2} & 0\\
0 & 0 & \frac{k+1}{2} & 0 & \frac{k-1}{2}\\
0 & 0 & 0 & k & 0
\end{pmatrix}
\]
[2605.25754].

## 3. Discrete curvature frameworks and exact or sharp estimates

Two curvature theories dominate the recent literature on amply regular graphs: Lin–Lu–Yau curvature and Bakry–Émery curvature. For \(p\in[0,1]\), the lazy measure at a vertex \(x\) is
\[
\mu_x^p = p\delta_x + (1-p)\frac{1}{k}\sum_{z\sim x}\delta_z
\]
in the \(k\)-regular case, and the \(p\)-Ollivier curvature is
\[
\kappa_p(x,y)=1-\frac{W_1(\mu_x^p,\mu_y^p)}{d(x,y)}.
\]
The Lin–Lu–Yau curvature is the derivative at \(p=1\),
\[
\kappa(x,y)=\lim_{p\to 1}\frac{\kappa_p(x,y)}{1-p},
\]
and for \(k\)-regular graphs one has the limit-free relation
\[
\kappa(x,y)=\frac{k+1}{k}\,\kappa_{1/(k+1)}(x,y)
\]
[2210.02988].

Early Hall-matching arguments gave exact or lower bounds in special regimes. For girth \(4\), equivalently \(\alpha=0\) and \(\beta\ge 2\), one has the exact formula
\[
\kappa(x,y)=\frac{2}{k}
\]
for all edges [2210.02988], [2111.04869]. In girth \(3\), Li–Liu established \(\kappa(x,y)=\mu/k\) when \(\lambda=1<\mu\), and lower bounds \(\kappa(x,y)\ge \mu/k\) when \(\lambda=\mu-1\) or \(\lambda=\mu>1\) [2111.04869]. In particular, every conference graph has positive Lin–Lu–Yau curvature; with parameters \((4t+1,2t,t-1,t)\), the Li–Liu bound yields \(\kappa(x,y)\ge 1/2\) [2111.04869].

Huang–Liu–Xia sharpened this in girth \(3\). If \(G\) is amply regular with parameters \((n,k,\alpha,\beta)\) and \(\beta>\alpha\ge 1\), then for every edge \(xy\),
\[
\kappa(x,y)\ge \frac{3}{k},
\]
and this is obtained by proving
\[
W_1\!\left(\mu_x^{1/(k+1)},\mu_y^{1/(k+1)}\right)\le \frac{k-2}{k+1}
\]
via a matching-induced transport plan on the regular bipartite graph \(H\) described above [2210.02988]. They also record the complementary upper bound
\[
\kappa(x,y)\le \frac{2+\alpha}{k},
\]
valid for any amply regular graph [2210.02988].

The 2024 paper “Ricci curvature, diameter and eigenvalues of amply regular graphs” improves the lower bound when \(1\neq \beta\ge \alpha\). For any edge \(xy\),
\[
\kappa_{\mathrm{LLY}}(x,y)\ge \frac{2+\left\lceil \alpha(\beta-\alpha)/(\beta-1)\right\rceil}{d},
\]
which recovers \(\kappa_{\mathrm{LLY}}=2/d\) for \(\alpha=0,\beta\ge 2\), improves the earlier \(3/d\) lower bound when \(\beta>\alpha\ge 1\), and approaches the upper bound \((2+\alpha)/d\) as \(\beta\to\infty\) [2410.21055]. The same paper proves that if \(\alpha>\beta\) and \(d\le \alpha+\beta\), then
\[
\kappa_{\mathrm{LLY}}(x,y)\ge \frac{2}{d}
\]
[2410.21055].

Conference graphs occupy a special place because their curvature can now be determined exactly. For a conference graph with parameters
\[
(v,k,\lambda,\mu)=(4\gamma+1,2\gamma,\gamma-1,\gamma),
\]
Chen–Liu–Zhang proved Bonini et al.’s conjecture:
\[
\kappa_{\mathrm{LLY}}(x,y)=\frac{\gamma+1}{2\gamma+1}
\]
for every edge \(xy\) [2409.06418]. Their method again proceeds by proving the existence of local perfect matchings, but the proof depends only on parameter relations and extends to broader amply regular families [2409.06418].

Bakry–Émery curvature enters through the curvature-dimension condition \(CD^\sigma(K,N)\). For unsigned graphs, the balanced-signature Bakry–Émery curvature at a vertex \(x\) in an amply regular graph satisfies
\[
K_{BE}(+,x)=2+\alpha/2+\left[\frac{2d(\beta-2)-\alpha^2}{2\beta}+\frac{2}{\beta}\min_{\lambda\in sp(A_{S_1(x)}|_{1^\perp})}\left(\lambda-\alpha/2\right)^2\right]_-,
\]
where \(A_{S_1(x)}\) is the adjacency matrix of the local graph induced on \(S_1(x)\) [2410.21055]. Consequences include \(K_{BE}(+,x)=2+\alpha/2\) whenever \(1\neq \beta\ge \alpha\) and \((\alpha,\beta)\neq (2,2)\), while the exceptional case \((2,2)\) depends on the local spectrum [2410.21055]. This formula has become a key input in finiteness, diameter, and connectivity arguments.

## 4. Diameter, eigenvalues, expansion, and edge-connectivity

Curvature estimates translate into global bounds through discrete Bonnet–Myers and Lichnerowicz-type inequalities. If \(\kappa(x,y)\ge \kappa_0>0\) on all edges, then
\[
\operatorname{diam}(G)\le \frac{2}{\kappa_0}
\]
[2210.02988]. Applying this to amply regular graphs yields several sharp consequences. Under \(1\neq \beta\ge \alpha\), one has \(\kappa(x,y)\ge 2/k\) and therefore \(\operatorname{diam}(G)\le k\); under \(\beta>\alpha\ge 1\), Huang–Liu–Xia obtain
\[
\operatorname{diam}(G)\le \left\lfloor \frac{2k}{3}\right\rfloor
\]
[2210.02988]. Their paper notes that for the \(9\)-Paley graph, the classical Neumaier–Penji diameter estimate gives \(\operatorname{diam}\le 4\), whereas \(\kappa_0=3/4\) yields \(\operatorname{diam}\le 2\), which is sharp [2210.02988].

The 2024 curvature paper strengthens these conclusions. If \(\beta\ge \max\{3,\alpha\}\), then
\[
\operatorname{diam}(G)\le
\left\lfloor
\frac{2d}{2+\max\left\{\alpha/2,\left\lceil ((\beta-\alpha)/(\beta-1))\alpha\right\rceil\right\}}
\right\rfloor
\]
[2410.21055]. It also proves a weak form of the Qiao–Park–Koolen conjecture: if \(\beta\ge \alpha>\varepsilon d\ge 1\), then \(\operatorname{diam}(G)\le 4/\varepsilon\) [2410.21055]. A complementary long-scale Ollivier approach gives further improvements for distance-regular graphs and then for amply regular graphs by combining Wasserstein contraction at scales \(1\) and \(q\ge 2\) [2412.18480].

Spectral consequences are equally direct. For a \(k\)-regular graph, with normalized Laplacian \(L=I-(1/k)A\), one has
\[
\lambda_1=1-\sigma_{n-1}/k,
\]
where \(\sigma_{n-1}\) is the second largest adjacency eigenvalue [2210.02988]. The discrete Lichnerowicz theorem gives
\[
\lambda_1\ge \inf_{xy\in E}\kappa(x,y),
\]
hence, under \(1\neq \beta\ge \alpha\),
\[
\sigma_{n-1}\le k-2,
\]
and, under \(\beta>\alpha\ge 1\),
\[
\sigma_{n-1}\le k-3
\]
[2210.02988]. The 2024 paper refines this to
\[
\theta_{n-1}\le d-2-\max\left\{\alpha/2,\left\lceil (\beta-\alpha)\alpha/(\beta-1)\right\rceil\right\}
\]
when \(1\neq \beta\ge \alpha\) and \((\alpha,\beta)\neq (2,2)\) [2410.21055]. It also derives lower bounds on \(\theta_1\), Cheeger and dual Cheeger estimates, an \((n,d,c)\)-expander conclusion, and a volume-growth inequality that is sharp for hypercubes [2410.21055].

Bakry–Émery curvature leads to connectivity statements of a different kind. Chen–Koolen–Liu prove that if a connected graph with minimum degree \(\delta\) satisfies \(CD(\infty,0)\), then it is \((\delta-1)\)-edge-connected [2507.18120]. For connected regular graphs with an even or infinite number of vertices, non-negative Bakry–Émery curvature then implies the existence of a perfect matching [2507.18120]. For amply regular graphs, the result is stronger: if \(G\) is connected amply regular with parameters \((d,\alpha,\beta)\) and \(\beta>1\), then
\[
\kappa'(G)=d.
\]
Moreover, if \(G\) is not a quadrangle, every minimum edge cut of size \(d\) consists of all \(d\) edges incident with a single vertex [2507.18120].

## 5. Classification results and rigidity phenomena

Several recent papers isolate parameter regimes in which amply regular graphs are completely classifiable. One such regime is Lichnerowicz sharpness. A graph is Lichnerowicz sharp if equality holds in the discrete Lichnerowicz bound, namely \(\lambda_1=\min_{xy\in E}\kappa(x,y)\) [2602.10396]. Chen–Liu–Zhang classify all Lichnerowicz sharp distance-regular graphs: they are precisely the cocktail party graphs \(CP(n)\), Hamming graphs \(H(d,n)\), Johnson graphs \(J(n,k)\), demi-cubes \(Q_{(2)}^n\), the Schlӓfli graph, and the Gosset graph [2602.10396]. Since non-complete distance-regular graphs are amply regular, this yields a complete classification of Lichnerowicz sharp amply regular graphs inside the distance-regular world [2602.10396].

The same paper classifies amply regular Terwilliger graphs with positive Lin–Lu–Yau curvature. Such a graph is isomorphic to exactly one of: the pentagon \(C_5\), the icosahedron, the line graph of the Petersen graph, the line graph of the Hoffman–Singleton graph, or the line graph of a strongly regular graph with parameters \((3250,57,0,1)\), contingent on existence [2602.10396]. The proof uses the edge-wise curvature inequality
\[
\kappa(x,y)\le \frac{2\alpha+3-d}{d},
\]
which implies that positive curvature forces \(d\le 2\alpha+2\), followed by a reduction through local strongly regular Terwilliger graphs [2602.10396]. A notable corollary is that no amply regular graph with \(\mu=1\) is Lichnerowicz sharp [2602.10396].

A different rigidity theorem, already noted above, treats the “near half valency” condition \(\mu=(k-1)/2\) for diameter at least \(4\). Jin–Koolen–Lv show that every connected amply regular graph in that regime is exactly one of \(Q_5\), \(\mathbf{K}_2\square \Lambda\), or the incidence graph of a \(GDDDP(2,k+1;\,k;\,0,\frac{k-1}{2})\) [2605.25754]. This places the extreme case just below the Brouwer–Cohen–Neumaier bound \(\mu\le k/2\) into a fully explicit framework involving bipartite \(Q\)-regular graphs, association schemes with five classes, and group divisible designs with the dual property [2605.25754].

Rigidity also appears in curvature extremality. If an amply regular graph satisfies
\[
2\beta-\alpha\ge k+1,
\]
then for every edge \(xy\),
\[
\kappa(x,y)=\frac{2+\alpha}{k},
\]
that is, the general curvature upper bound is attained [2210.02988]. The proof uses a perfect matching in the bipartite graph between \(N_x\) and \(N_y\) induced by the original graph [2210.02988]. Johnson graph \(J(4,2)\), with parameters \((6,4,2,4)\), satisfies this and has \(\kappa(x,y)=1\) on every edge [2210.02988].

These results also clarify common misconceptions. A frequent expectation is that curvature, or even positive curvature, should be determined solely by the parameter quadruple \((v,k,\lambda,\mu)\). This fails in girth \(3\): the Shrikhande graph and the \(4\times 4\) Rook’s graph share parameters \((16,6,2,2)\), yet have distinct curvature values, \(1/3\) and \(2/3\) respectively [2210.02988]. A plausible implication is that local induced subgraph structure and matching data must be included in any finer classification of geometric behavior.

## 6. Constructions, clique bounds, and directions of current research

Constructive sources of amply regular graphs come from both algebraic combinatorics and finite-field harmonic analysis. The broadest explicit construction in the supplied material comes from weakly regular \(p\)-ary plateaued functions over finite fields of odd characteristic. Feng, Li, Mesnager, and collaborators construct strongly regular Cayley graphs from the partial difference sets
\[
D_f,\quad D_{f,sq},\quad D_{f,nsq},\quad D_{f,sq,0},
\]
obtained from a weakly regular \(s\)-plateaued function satisfying \(f(0)=0\), a homogeneity condition \(f(ax)=a^h f(x)\), and the parity condition \(n+s\) even [1809.03422]. Since every strongly regular graph is amply regular, these yield families of amply regular graphs of diameter \(2\) with explicit parameter sets of three types [1809.03422]. The same paper also constructs symmetric association schemes of class \(p\), suggesting a route from plateaued functions to broader combinatorial structures closely linked to amply regularity [1809.03422].

Design-theoretic constructions appear in the classification of the \(\mu=(k-1)/2\) regime. The incidence graph of a \(GDDDP(2,k+1;\,k;\,0,\frac{k-1}{2})\) is bipartite, \(k\)-regular, triangle-free, and amply regular with parameters
\[
(v,k,\lambda,\mu)=\left(4k+4,\;k,\;0,\;\frac{k-1}{2}\right)
\]
[2605.25754]. Infinite families arise from Paley graphs, Peisert graphs, and Paley digraphs via Taylor extension, bipartite double, or extended double cover constructions, all producing graphs with
\[
(v,k,\lambda,\mu)=\left(4(q+1),\;q,\;0,\;\frac{q-1}{2}\right),
\qquad d=4
\]
[2605.25754].

Clique geometry provides another active direction. Gavrilyuk and Koolen study maximal cliques in amply regular graphs with fixed smallest eigenvalue \(s=-m\). If \(C\) is a maximal clique of size \(c\), under the assumptions \(\mu>m(m-1)\) and
\[
c>\frac{\mu-m(m-1)}{2}-m+1,
\]
they derive the cubic inequality
\[
M_\Gamma(c)\ge 0,
\]
where
\[
M_\Gamma(c)=\left[(c+m-3)(k-c+1)-2(c-1)(\lambda-c+2)\right]^2
-(k-c+1)^2(c+m-1)\bigl(c-(m-1)(4m-1)\bigr)
\]
[2012.09391]. Since \(M_\Gamma\) is a cubic polynomial in \(c\) with positive leading coefficient, maximal cliques are forced into a “small or large” dichotomy [2012.09391]. In strongly regular graphs containing a Delsarte clique, the same method implies that \(\mu\) is either small or large according to an explicit square-root bound [2012.09391]. These results have been used to rule out infinite families of feasible strongly regular parameter sets [2012.09391].

Current open directions, as explicitly identified in the supplied papers, include determining finer curvature bounds for girth \(3\) when \(\beta\le \alpha\), extending matching-based transport methods beyond the \(\beta>\alpha\) regime, refining the Qiao–Park–Koolen conjecture on diameter, approaching Terwilliger’s finiteness conjecture \(\beta\ge 2\), and deciding whether the Paley-, Peisert-, and Paley-digraph-based constructions exhaust the \(GDDDP(2,q+1;\,q;\,0,\frac{q-1}{2})\) family [2210.02988], [2410.21055], [2605.25754]. Another recurring theme is that positive curvature in amply regular graphs appears to be highly restrictive, but the full boundary between positive and non-positive curvature remains unsettled outside the classified Terwilliger and conference cases [2602.10396], [2409.06418], [2507.18120].

Source: https://www.emergentmind.com/topics/amply-regular-graphs