Triple intersection numbers for the Paley graphs

We give a tight bound for the triple intersection numbers of Paley graphs. In particular, we show that any three vertices have a common neighbor in Paley graphs of order larger than 25.     

Paley type partial difference sets in abelian groups

Partial difference sets with parameters $\left(v,k,\lambda ,\mu \right)=\left(v,\left(v?1\right)/2,\left(v?5\right)/4,\left(v?1\right)/4\right)$ are called Paley type partial difference sets. In this note, we prove that if there exists a Paley type partial difference set in an abelian group of order v, where v is not a prime power, then $v=n^4$ or $9n^4$, $n>1$ an odd integer. In 2010, Polhill constructed Paley type partial difference sets in abelian groups with those orders. Thus, combining with the constructions of Polhill and the classical Paley construction using nonzero squares of a finite field, we completely answer the following question: “For which odd positive integers $v>1$, can we find a Paley type partial difference set in an abelian group of order $v$?”     

Some implications on amorphic association schemes

We give an overview of results on amorphic association schemes. We give the known constructions of such association schemes, and enumerate most such association schemes on up to 49 vertices. Special attention is paid to cyclotomic association schemes. We give several results on when a strongly regular decomposition of the complete graph is an amorphic association scheme. This includes a new proof of the result that a decomposition of the complete graph into three strongly regular graphs is an amorphic association scheme, and the new result that a strongly regular decomposition of the complete graph for which the union of any two relations is again strongly regular must be an amorphic association scheme.     

New negative Latin square type partial difference sets in nonelementary abelian 2-groups and 3-groups

A partial difference set having parameters (n ², r(n − 1), n + r ² − 3r, r ² − r) is called a Latin square type partial difference set, while a partial difference set having parameters (n ², r(n + 1), − n + r ² + 3r, r ² + r) is called a negative Latin square type partial difference set. Nearly all known constructions of negative Latin square partial difference sets are in elementary abelian groups. In this paper, we develop three product theorems that construct negative Latin square type partial difference sets and Latin square type partial difference sets in direct products of abelian groups G and G′ when these groups have certain Latin square or negative Latin square type partial difference sets. Using these product theorems, we can construct negative Latin square type partial difference sets in groups of the form where the s _i are nonnegative integers and s ₀ + s ₁ ≥ 1. Another significant corollary to these theorems are constructions of two infinite families of negative Latin square type partial difference sets in 3-groups of the form for nonnegative integers s _i . Several constructions of Latin square type PDSs are also given in p-groups for all primes p. We will then briefly indicate how some of these results relate to amorphic association schemes. In particular, we construct amorphic association schemes with 4 classes using negative Latin square type graphs in many nonelementary abelian 2-groups; we also use negative Latin square type graphs whose underlying sets can be elementary abelian 3-groups or nonelementary abelian 3-groups to form 3-class amorphic association schemes.      

Amorphic association schemes from bent partitions

Bent partitions are partitions of an elementary abelian group, which have similarities to partitions from spreads. In fact, a spread partition is a special case of a bent partition. In particular, bent partitions give rise to a large number of (vectorial) bent functions. Examples of bent partitions, which generalize the Desarguesian spread, have been presented by Anbar, Meidl and Pirsic, 2021, 2022. Bent partitions, which generalize some other classes of (pre)semifield spreads, have been presented by Anbar, Kalaycı, Meidl 2023. In this article, it is shown that these bent partitions induce $(p^k+1)$-class amorphic associations schemes on ${\mathbb{F}}_{p^m}\times {\mathbb{F}}_{p^m}$, where k is a divisor of m with special properties. This implies a construction of amorphic association schemes from some classes of (pre)semifields.     

Negative Latin square type partial difference sets and amorphic association schemes with Galois rings

A partial difference set (PDS) having parameters (n², r(n?1), n+r²?3r, r²?r) is called a Latin square type PDS, while a PDS having parameters (n², r(n+1), ?n+r²+3r, r² +r) is called a negative Latin square type PDS. There are relatively few known constructions of negative Latin square type PDSs, and nearly all of these are in elementary abelian groups. We show that there are three different groups of order 256 that have all possible negative Latin square type parameters. We then give generalized constructions of negative Latin square type PDSs in 2‐groups. We conclude by discussing how these results fit into the context of amorphic association schemes and by stating some open problems. © 2009 Wiley Periodicals, Inc. J Combin Designs 17: 266‐282, 2009     

Strongly regular Cayley graphs from partitions of subdifference sets of the Singer difference sets

In this paper, we give a new lifting construction of “hyperbolic” type of strongly regular Cayley graphs. Also we give new constructions of strongly regular Cayley graphs over the additive groups of finite fields based on partitions of subdifference sets of the Singer difference sets. Our results unify some recent constructions of strongly regular Cayley graphs related to m-ovoids and i-tight sets in finite geometry. Furthermore, some of the strongly regular Cayley graphs obtained in this paper are new or nonisomorphic to known strongly regular graphs with the same parameters.     

On a family of highly regular graphs by Brouwer,Ivanov, and Klin

Highly regular graphs for which not all regularities are explainable by symmetries are fascinating creatures. Some of them like, e.g., the line graph of W. Kantor’s non-classical $GQ(5^2,5)$, are stumbling stones for existing implementations of graph isomorphism tests. They appear to be extremely rare and even once constructed it is difficult to prove their high regularity. Yet some of them, like the McLaughlin graph on 275 vertices and Ivanov’s graph on 256 vertices are of profound beauty. This alone makes it an attractive goal to strive for their complete classification or, failing this, at least to get a deep understanding of them. Recently, one of the authors discovered new methods for proving high regularity of graphs. Using these techniques, in this paper we study a classical family of strongly regular graphs, originally discovered by A.E. Brouwer, A.V. Ivanov, and M.H. Klin in the late 80s. We analyse their symmetries and show that they are $(3,5)$-regular but not 2-homogeneous. Thus we promote these graphs to the distinguished club of highly regular graphs with few symmetries.     

Eigenvalues of association schemes of quadratic forms

The duality and primitivity of the association scheme Qua(n,q) of quadratic forms in n variables and the association scheme Sym(n,q) of symmetric bilinear forms in n variables over the finite field are discussed by Wang et al. [Association schemes of quadratic forms and symmetric bilinear forms, J. Algebraic Combin. 17 (2003) 149–161]. In this paper, eigenvalues of Qua(n,q) are computed, where q is a power of 2. As an application, two fusion schemes of Qua(n,q) are discussed and their dual schemes are constructed.     

Lifting constructions of strongly regular Cayley graphs

We give two “lifting” constructions of strongly regular Cayley graphs. In the first construction we “lift” a cyclotomic strongly regular graph by using a subdifference set of the Singer difference sets. The second construction uses quadratic forms over finite fields and it is a common generalization of the construction of the affine polar graphs [7] and a construction of strongly regular Cayley graphs given in [15]. The two constructions are related in the following way: the second construction can be viewed as a recursive construction, and the strongly regular Cayley graphs obtained from the first construction can serve as starters for the second construction. We also obtain association schemes from the second construction.     

Strongly regular graphs associated with ternary bent functions

We prove a new characterization of weakly regular ternary bent functions via partial difference sets. Partial difference sets are combinatorial objects corresponding to strongly regular graphs. Using known families of bent functions, we obtain in this way new families of strongly regular graphs, some of which were previously unknown. One of the families includes an example in [N. Hamada, T. Helleseth, A characterization of some {3v₂+v₃,3v₁+v₂,3,3}-minihypers and some [15,4,9;3]-codes with B₂=0, J. Statist. Plann. Inference 56 (1996) 129-146], which was considered to be sporadic; using our results, this strongly regular graph is now a member of an infinite family. Moreover, this paper contains a new proof that the Coulter-Matthews and ternary quadratic bent functions are weakly regular.     

Finite Euclidean graphs and Ramanujan graphs

We consider finite analogues of Euclidean graphs in a more general setting than that considered in [A. Medrano, P. Myers, H.M. Stark, A. Terras, Finite analogues of Euclidean space, J. Comput. Appl. Math. 68 (1996) 221-238] and we obtain many new examples of Ramanujan graphs. In order to prove these results, we use the previous work of [W.M. Kwok, Character tables of association schemes of affine type, European J. Combin. 13 (1992) 167-185] calculating the character tables of certain association schemes of affine type. A key observation is that we can obtain better estimates for the ordinary Kloosterman sum K(a,b;q). In particular, we always achieve , and in many (but not all) of the cases, instead of the well known . Also, we use the ideas of controlling association schemes, and the Ennola type dualities, in our previous work on the character tables of commutative association schemes. The method in this paper will be used to construct many more new examples of families of Ramanujan graphs in the subsequent paper.     

Partial difference sets and amorphic Cayley schemes in non‐abelian 2‐groups

In this paper, we consider regular automorphism groups of graphs in the RT2 family and the Davis‐Xiang family and amorphic abelian Cayley schemes from these graphs. We derive general results on the existence of non‐abelian regular automorphism groups from abelian regular automorphism groups and apply them to the RT2 family and Davis‐Xiang family and their amorphic abelian Cayley schemes to produce amorphic non‐abelian Cayley schemes.     

Negative Latin square type partial difference sets in nonabelian groups of order 64

There exist few examples of negative Latin square type partial difference sets (NLST PDSs) in nonabelian groups. We present a list of 176 inequivalent NLST PDSs in 48 nonisomorphic, nonabelian groups of order 64. These NLST PDSs form 8 nonisomorphic strongly regular graphs. These PDSs were constructed using a combination of theoretical techniques and computer search, both of which are described. The search was run exhaustively on 212/267 nonisomorphic groups of order 64.     

A hemisystem of a nonclassical generalised quadrangle

The concept of a hemisystem of a generalised quadrangle has its roots in the work of B. Segre, and this term is used here to denote a set of points such that every line ℓ meets in half of the points of ℓ. If one takes the point-line geometry on the points of the hemisystem, then one obtains a partial quadrangle and hence a strongly regular point graph. The only previously known hemisystems of generalised quadrangles of order (q, q ²) were those of the elliptic quadric , q odd. We show in this paper that there exists a hemisystem of the Fisher–Thas–Walker–Kantor generalised quadrangle of order (5, 5²), which leads to a new partial quadrangle. Moreover, we can construct from our hemisystem the 3· A ₇-hemisystem of , first constructed by Cossidente and Penttila.      

Investigations on association schemes with elementary abelian thin residue

In this paper, we give a new class of association schemes whose thin residues are isomorphic to an elementary abelian $p$-group of order $p^2$. We then study the automorphism groups of these schemes and determine whether these schemes are schurian.     

Using equality in the Krein conditions to prove nonexistence of certain distance-regular graphs

We prove the nonexistence of a distance-regular graph with intersection array {74,54,15;1,9,60} and of distance-regular graphs with intersection arrays

{4r³+8r²+6r+1,2r(r+1)(2r+1),2r²+2r+1;1,2r(r+1),(2r+1)(2r²+2r+1)}