Amply Regular Graphs

• Amply regular graphs are connected k-regular graphs defined by parameters (v, k, λ, μ) that fix the number of common neighbors for adjacent and distance-2 vertex pairs, generalizing strongly regular graphs.
• Their structure facilitates analysis using discrete curvature methods, matching-based transport, and spectral estimates to derive diameter bounds, expansion rates, and edge-connectivity properties.
• Recent research employs constructions from designs, finite-field functions, and association schemes to classify these graphs and explore rigidity and curvature phenomena.

Searching arXiv for 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 $k=b_0$, $2$0, and $2$1 (Huang et al., 2022, Chen et al., 11 Feb 2026). 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 (Chen et al., 2024, Chen et al., 24 Jul 2025, Mesnager et al., 2018).

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

A graph $2$2 on $2$3 vertices is called amply regular with parameters $2$4 if $2$5 is $2$6-regular, any two adjacent vertices have exactly $2$7 common neighbors, and any two vertices at distance $2$8 have exactly $2$9 common neighbors (Huang et al., 2022). In the alternative notation $(v,k,\lambda,\mu)$0, one writes $(v,k,\lambda,\mu)$1, $(v,k,\lambda,\mu)$2, $(v,k,\lambda,\mu)$3, and $(v,k,\lambda,\mu)$4 (Chen et al., 11 Feb 2026). The class is local in nature: its defining conditions constrain only spheres of radius at most $(v,k,\lambda,\mu)$5, and therefore amply regular graphs may have diameter greater than $(v,k,\lambda,\mu)$6, finite or infinite order, and a wide range of global structures (Chen et al., 2024, Chen et al., 24 Jul 2025).

Strongly regular graphs are precisely the amply regular graphs of diameter $(v,k,\lambda,\mu)$7; for them, “non-adjacent” and “distance $(v,k,\lambda,\mu)$8” coincide (Mesnager et al., 2018). Every non-complete distance-regular graph is amply regular, with parameter identification $(v,k,\lambda,\mu)$9 (Chen et al., 11 Feb 2026). Terwilliger graphs form a distinguished subclass: a non-complete graph is a Terwilliger graph if for any vertices $k$0 at distance $k$1, the common neighbors of $k$2 and $k$3 induce a clique of size $k$4; in amply regular graphs this is equivalent to having no induced quadrangles (Chen et al., 11 Feb 2026).

Standard local notation around an edge $k$5 is central. Writing $k$6 for the neighborhood of $k$7, one sets

$k$8

and

$k$9

Then $v$0 and $v$1 (Huang et al., 2022). These sets drive most modern arguments on curvature and matching.

A basic universal inequality relates the parameters: $v$2 with equality if and only if the graph is strongly regular (Chen et al., 2024). Additional structural inequalities appear when the diameter is at least $v$3 or $v$4. For amply regular graphs with diameter $v$5, one has $v$6, with equality if and only if the graph is either a polygon or a Hadamard graph; also $v$7, hence $v$8 (Jin et al., 25 May 2026). Other bounds used in curvature-based analyses include

$v$9

with equality iff the graph is the icosahedron or a line graph of a regular graph of girth at least $\lambda$0, and, if $\lambda$1,

$\lambda$2

(Chen et al., 2024).

2. Local combinatorics, matching structure, and neighborhood geometry

Much of the modern theory proceeds by encoding the local combinatorics around an edge $\lambda$3 into auxiliary bipartite graphs whose perfect matchings control transport and curvature. In the girth-$\lambda$4 case with $\lambda$5, a transport-bipartite graph $\lambda$6 can be built from $\lambda$7, $\lambda$8, the common-neighbor set $\lambda$9, a copy $2$0, and balancing vertices $2$1 when $2$2 (Huang et al., 2022). Its edge set consists of eight explicitly defined families $2$3, and one checks that every vertex has degree $2$4; thus $2$5 is $2$6-regular and decomposes into $2$7 edge-disjoint perfect matchings by Kőnig’s theorem (Huang et al., 2022).

A different but related perspective appears in the study of sharp curvature upper bounds. For an edge $2$8 in a $2$9-regular graph, the Lin–Lu–Yau curvature satisfies

$\mu$0

and equality holds iff there exists a perfect matching between $\mu$1 and $\mu$2 (Chen et al., 2024). 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

$\mu$3

where $\mu$4 and $\mu$5 (Chen et al., 2024). The key observation is that common-neighbor counts yield quadratic inequalities in the total number of edges from a Hall-obstruction subset $\mu$6 into $\mu$7, and these inequalities rule out Hall failure under explicit parameter regimes (Chen et al., 2024).

This matching viewpoint also clarifies why local parameters alone do not always determine global geometry. In the strongly regular parameter set $\mu$8, the Shrikhande graph and the $\mu$9 Rook’s graph have different curvature values, even though their $2$0 parameters coincide (Huang et al., 2022). This suggests that local matching structure inside and around $2$1, rather than the parameter quadruple alone, can be decisive.

A further rigidity phenomenon arises in the regime $2$2 with diameter at least $2$3. Jin–Koolen–Lv prove that connected amply regular graphs with $2$4 odd, $2$5, and

$2$6

must satisfy $2$7 and $2$8, and are then exactly one of three types: the $2$9-cube, the graph $k=b_0$0 where $k=b_0$1 is the unique bipartite $k=b_0$2-graph on $k=b_0$3 vertices, or the point-block incidence graph of a $k=b_0$4 (Jin et al., 25 May 2026). In the last case the graph is bipartite, has diameter $k=b_0$5, and admits an equitable distance partition with quotient matrix

$k=b_0$6

(Jin et al., 25 May 2026).

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 $k=b_0$7, the lazy measure at a vertex $k=b_0$8 is

$k=b_0$9

in the $2$00-regular case, and the $2$01-Ollivier curvature is

$2$02

The Lin–Lu–Yau curvature is the derivative at $2$03,

$2$04

and for $2$05-regular graphs one has the limit-free relation

$2$06

(Huang et al., 2022).

Early Hall-matching arguments gave exact or lower bounds in special regimes. For girth $2$07, equivalently $2$08 and $2$09, one has the exact formula

$2$10

for all edges (Huang et al., 2022, Li et al., 2021). In girth $2$11, Li–Liu established $2$12 when $2$13, and lower bounds $2$14 when $2$15 or $2$16 (Li et al., 2021). In particular, every conference graph has positive Lin–Lu–Yau curvature; with parameters $2$17, the Li–Liu bound yields $2$18 (Li et al., 2021).

Huang–Liu–Xia sharpened this in girth $2$19. If $2$20 is amply regular with parameters $2$21 and $2$22, then for every edge $2$23,

$2$24

and this is obtained by proving

$2$25

via a matching-induced transport plan on the regular bipartite graph $2$26 described above (Huang et al., 2022). They also record the complementary upper bound

$2$27

valid for any amply regular graph (Huang et al., 2022).

The 2024 paper “Ricci curvature, diameter and eigenvalues of amply regular graphs” improves the lower bound when $2$28. For any edge $2$29,

$2$30

which recovers $2$31 for $2$32, improves the earlier $2$33 lower bound when $2$34, and approaches the upper bound $2$35 as $2$36 (Chen et al., 2024). The same paper proves that if $2$37 and $2$38, then

$2$39

(Chen et al., 2024).

Conference graphs occupy a special place because their curvature can now be determined exactly. For a conference graph with parameters

$2$40

Chen–Liu–Zhang proved Bonini et al.’s conjecture: $2$41 for every edge $2$42 (Chen et al., 2024). 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 (Chen et al., 2024).

Bakry–Émery curvature enters through the curvature-dimension condition $2$43. For unsigned graphs, the balanced-signature Bakry–Émery curvature at a vertex $2$44 in an amply regular graph satisfies

$2$45

where $2$46 is the adjacency matrix of the local graph induced on $2$47 (Chen et al., 2024). Consequences include $2$48 whenever $2$49 and $2$50, while the exceptional case $2$51 depends on the local spectrum (Chen et al., 2024). 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 $2$52 on all edges, then

$2$53

(Huang et al., 2022). Applying this to amply regular graphs yields several sharp consequences. Under $2$54, one has $2$55 and therefore $2$56; under $2$57, Huang–Liu–Xia obtain

$2$58

(Huang et al., 2022). Their paper notes that for the $2$59-Paley graph, the classical Neumaier–Penji diameter estimate gives $2$60, whereas $2$61 yields $2$62, which is sharp (Huang et al., 2022).

The 2024 curvature paper strengthens these conclusions. If $2$63, then

$2$64

(Chen et al., 2024). It also proves a weak form of the Qiao–Park–Koolen conjecture: if $2$65, then $2$66 (Chen et al., 2024). 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 $2$67 and $2$68 (Chen et al., 2024).

Spectral consequences are equally direct. For a $2$69-regular graph, with normalized Laplacian $2$70, one has

$2$71

where $2$72 is the second largest adjacency eigenvalue (Huang et al., 2022). The discrete Lichnerowicz theorem gives

$2$73

hence, under $2$74,

$2$75

and, under $2$76,

$2$77

(Huang et al., 2022). The 2024 paper refines this to

$2$78

when $2$79 and $2$80 (Chen et al., 2024). It also derives lower bounds on $2$81, Cheeger and dual Cheeger estimates, an $2$82-expander conclusion, and a volume-growth inequality that is sharp for hypercubes (Chen et al., 2024).

Bakry–Émery curvature leads to connectivity statements of a different kind. Chen–Koolen–Liu prove that if a connected graph with minimum degree $2$83 satisfies $2$84, then it is $2$85-edge-connected (Chen et al., 24 Jul 2025). 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 (Chen et al., 24 Jul 2025). For amply regular graphs, the result is stronger: if $2$86 is connected amply regular with parameters $2$87 and $2$88, then

$2$89

Moreover, if $2$90 is not a quadrangle, every minimum edge cut of size $2$91 consists of all $2$92 edges incident with a single vertex (Chen et al., 24 Jul 2025).

5. Classification results and rigidity phenomena

Several 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 $2$93 (Chen et al., 11 Feb 2026). Chen–Liu–Zhang classify all Lichnerowicz sharp distance-regular graphs: they are precisely the cocktail party graphs $2$94, Hamming graphs $2$95, Johnson graphs $2$96, demi-cubes $2$97, the Schlӓfli graph, and the Gosset graph (Chen et al., 11 Feb 2026). 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 (Chen et al., 11 Feb 2026).

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 $2$98, 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 $2$99, contingent on existence (Chen et al., 11 Feb 2026). The proof uses the edge-wise curvature inequality

$(v,k,\lambda,\mu)$00

which implies that positive curvature forces $(v,k,\lambda,\mu)$01, followed by a reduction through local strongly regular Terwilliger graphs (Chen et al., 11 Feb 2026). A notable corollary is that no amply regular graph with $(v,k,\lambda,\mu)$02 is Lichnerowicz sharp (Chen et al., 11 Feb 2026).

A different rigidity theorem, already noted above, treats the “near half valency” condition $(v,k,\lambda,\mu)$03 for diameter at least $(v,k,\lambda,\mu)$04. Jin–Koolen–Lv show that every connected amply regular graph in that regime is exactly one of $(v,k,\lambda,\mu)$05, $(v,k,\lambda,\mu)$06, or the incidence graph of a $(v,k,\lambda,\mu)$07 (Jin et al., 25 May 2026). This places the extreme case just below the Brouwer–Cohen–Neumaier bound $(v,k,\lambda,\mu)$08 into a fully explicit framework involving bipartite $(v,k,\lambda,\mu)$09-regular graphs, association schemes with five classes, and group divisible designs with the dual property (Jin et al., 25 May 2026).

Rigidity also appears in curvature extremality. If an amply regular graph satisfies

$(v,k,\lambda,\mu)$10

then for every edge $(v,k,\lambda,\mu)$11,

$(v,k,\lambda,\mu)$12

that is, the general curvature upper bound is attained (Huang et al., 2022). The proof uses a perfect matching in the bipartite graph between $(v,k,\lambda,\mu)$13 and $(v,k,\lambda,\mu)$14 induced by the original graph (Huang et al., 2022). Johnson graph $(v,k,\lambda,\mu)$15, with parameters $(v,k,\lambda,\mu)$16, satisfies this and has $(v,k,\lambda,\mu)$17 on every edge (Huang et al., 2022).

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)$18. This fails in girth $(v,k,\lambda,\mu)$19: the Shrikhande graph and the $(v,k,\lambda,\mu)$20 Rook’s graph share parameters $(v,k,\lambda,\mu)$21, yet have distinct curvature values, $(v,k,\lambda,\mu)$22 and $(v,k,\lambda,\mu)$23 respectively (Huang et al., 2022). 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 $(v,k,\lambda,\mu)$24-ary plateaued functions over finite fields of odd characteristic. Feng, Li, Mesnager, and collaborators construct strongly regular Cayley graphs from the partial difference sets

$(v,k,\lambda,\mu)$25

obtained from a weakly regular $(v,k,\lambda,\mu)$26-plateaued function satisfying $(v,k,\lambda,\mu)$27, a homogeneity condition $(v,k,\lambda,\mu)$28, and the parity condition $(v,k,\lambda,\mu)$29 even (Mesnager et al., 2018). Since every strongly regular graph is amply regular, these yield families of amply regular graphs of diameter $(v,k,\lambda,\mu)$30 with explicit parameter sets of three types (Mesnager et al., 2018). The same paper also constructs symmetric association schemes of class $(v,k,\lambda,\mu)$31, suggesting a route from plateaued functions to broader combinatorial structures closely linked to amply regularity (Mesnager et al., 2018).

Design-theoretic constructions appear in the classification of the $(v,k,\lambda,\mu)$32 regime. The incidence graph of a $(v,k,\lambda,\mu)$33 is bipartite, $(v,k,\lambda,\mu)$34-regular, triangle-free, and amply regular with parameters

$(v,k,\lambda,\mu)$35

(Jin et al., 25 May 2026). 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)$36

(Jin et al., 25 May 2026).

Clique geometry provides another active direction. Gavrilyuk and Koolen study maximal cliques in amply regular graphs with fixed smallest eigenvalue $(v,k,\lambda,\mu)$37. If $(v,k,\lambda,\mu)$38 is a maximal clique of size $(v,k,\lambda,\mu)$39, under the assumptions $(v,k,\lambda,\mu)$40 and

$(v,k,\lambda,\mu)$41

they derive the cubic inequality

$(v,k,\lambda,\mu)$42

where

$(v,k,\lambda,\mu)$43

(2012.09391). Since $(v,k,\lambda,\mu)$44 is a cubic polynomial in $(v,k,\lambda,\mu)$45 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 $(v,k,\lambda,\mu)$46 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 $(v,k,\lambda,\mu)$47 when $(v,k,\lambda,\mu)$48, extending matching-based transport methods beyond the $(v,k,\lambda,\mu)$49 regime, refining the Qiao–Park–Koolen conjecture on diameter, approaching Terwilliger’s finiteness conjecture $(v,k,\lambda,\mu)$50, and deciding whether the Paley-, Peisert-, and Paley-digraph-based constructions exhaust the $(v,k,\lambda,\mu)$51 family (Huang et al., 2022, Chen et al., 2024, Jin et al., 25 May 2026). 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 (Chen et al., 11 Feb 2026, Chen et al., 2024, Chen et al., 24 Jul 2025).