A class of amply regular graphs related to the subconstituents of a dual polar graph

The connected components of the induced graphs on each subconstituent of the dual polar graph of the odd dimensional orthogonal spaces over a finite field are shown to be amply regular. The connected components of the graphs on the second and third subconstituents are shown to be distance-regular by elementary methods.     

Distance regular graphs arising from dimensional dual hyperovals

In [12], A. Pasini and S. Yoshiara studied the distance regular graphs constructed from the Yoshiara dual hyperovals. In this note, we prove that the incidence graphs of the semibiplanes constructed from dimensional dual hyperovals are distance regular graphs if the dual hyperovals are doubly dual hyperovals (DDHOs). This generalizes the result in [12].     

Weak Rickart and dual weak Rickart objects in abelian categories

We introduce and investigate weak relative Rickart objects and dual weak relative Rickart objects in abelian categories. Several types of abelian categories are characterized in terms of (dual) weak relative Rickart properties. We relate our theory to the study of relative regular objects and (dual) relative Baer objects. We also give some applications to module and comodule categories.     

Eulerian partial duals of plane graphs

In the paper Combinatorica 33(2) (2013) 231–252, Huggett and Moffatt characterized all bipartite partial duals of a plane graph in terms of oriented circuits in its medial graph. An open problem posed in their paper is the characterization of Eulerian partial duals of plane graphs. In this article, we solve this problem by considering half‐edge orientations of medial graphs.     

Metric characterization of apartments in dual polar spaces

Let Π be a polar space of rank n and let Gk(Π), k∈{0,…,n−1} be the polar Grassmannian formed by k-dimensional singular subspaces of Π. The corresponding Grassmann graph will be denoted by Γk(Π). We consider the polar Grassmannian Gn−1(Π) formed by maximal singular subspaces of Π and show that the image of every isometric embedding of the n-dimensional hypercube graph Hn in Γn−1(Π) is an apartment of Gn−1(Π). This follows from a more general result concerning isometric embeddings of Hm, m?n in Γn−1(Π). As an application, we classify all isometric embeddings of Γn−1(Π) in Γn^′−1(Π^′), where Π^′ is a polar space of rank n^′?n.     

Deza graphs with parameters and

A Deza graph with parameters is a ‐regular graph with vertices, in which any two vertices have or () common neighbours. A Deza graph is strictly Deza if it has diameter , and is not strongly regular. In an earlier paper, the two last authors et al characterised the strictly Deza graphs with and , where is the number of vertices with common neighbours with a given vertex. Here, we start with a characterisation of Deza graphs (not necessarily strictly Deza graphs) with parameters . Then, we deal with the case and , and thus complete the characterisation of Deza graphs with . It follows that all Deza graphs with , and can be made from special strongly regular graphs, and in fact are strictly Deza except for . We present several examples of such strongly regular graphs. A divisible design graph (DDG) is a special Deza graph, and a Deza graph with is a DDG. The present characterisation reveals an error in a paper on DDGs by the second author et al. We discuss the cause and the consequences of this mistake and give the required errata.     

On CCE graphs of doubly partial orders

Let D be a digraph. The competition-common enemy graph (CCE graph) of D has the same set of vertices as D and an edge between vertices u and v if and only if there are vertices w and x in D such that (w,u), (w,v), (u,x), and (v,x) are arcs of D. We call a graph a CCE graph if it is the CCE graph of some digraph. In this paper, we show that if the CCE graph of a doubly partial order does not contain C₄ as an induced subgraph, it is an interval graph. We also show that any interval graph together with enough isolated vertices is the CCE graph of some doubly partial order.     

Intriguing sets in partial quadrangles

The point‐line geometry known as a partial quadrangle (introduced by Cameron in 1975) has the property that for every point/line non‐incident pair (P, ?), there is at most one line through P concurrent with ?. So in particular, the well‐studied objects known as generalized quadrangles are each partial quadrangles. An intriguing set of a generalized quadrangle is a set of points which induces an equitable partition of size two of the underlying strongly regular graph. We extend the theory of intriguing sets of generalized quadrangles by Bamberg, Law and Penttila to partial quadrangles, which gives insight into the structure of hemisystems and other intriguing sets of generalized quadrangles. © 2010 Wiley Periodicals, Inc. J Combin Designs 19:217‐245, 2011     

Restrictions on parameters of partial difference sets in nonabelian groups

A partial difference set $S$ in a finite group $G$ satisfying $1?S$ and $S=S^{?1}$ corresponds to an undirected strongly regular Cayley graph $\text{Cay}\left(G,S\right)$. While the case when $G$ is abelian has been thoroughly studied, there are comparatively few results when $G$ is nonabelian. In this paper, we provide restrictions on the parameters of a partial difference set that apply to both abelian and nonabelian groups and are especially effective in groups with a nontrivial center. In particular, these results apply to $p$‐groups, and we are able to rule out the existence of partial difference sets in many instances.     

Characterizations of bipartite and Eulerian partial duals of ribbon graphs

Huggett and Moffatt characterized all bipartite partial duals of a plane graph in terms of all-crossing directions of its medial graph. Then Metsidik and Jin characterized all Eulerian partial duals of a plane graph in terms of semi-crossing directions of its medial graph. Plane graphs are ribbon graphs with genus 0. In this paper, by introducing the notion of modified medial graphs and using their all-crossing directions, we first extend Huggett and Moffatt’s result from plane graphs to ribbon graphs. Then we characterize all Eulerian partial duals of any ribbon graph in terms of crossing-total directions of its medial graph, which are simpler than semi-crossing directions.     

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 $v$?”     

Classification of subsets with minimal width and dual width in Grassmann, bilinear forms and dual polar graphs

Brouwer, Godsil, Koolen and Martin [Width and dual width of subsets in polynomial association schemes, J. Combin. Theory Ser. A 102 (2003) 255-271] introduced the width w and the dual width w^* of a subset in a distance-regular graph and in a cometric association scheme, respectively, and then derived lower bounds on these new parameters. For instance, subsets with the property w+w^*=d in a cometric distance-regular graph with diameter d attain these bounds. In this paper, we classify subsets with this property in Grassmann graphs, bilinear forms graphs and dual polar graphs. We use this information to establish the Erd?s-Ko-Rado theorem in full generality for the first two families of graphs.     

关于PFI-代数与剩余格

本文提出了一种强FI代数-PFI代数,并且深入研究了它的性质,借此进一步揭示了FI-代数和剩余格之间更加密切的联系,进而以FI-代数为基本框架建立了R0-代数、正则剩余格等逻辑系统的结构特征(包括对隅结构)及其相互关系．这种以FI-代数为基础来统一处理剩余格和R0-代数的方法,同样适合于格蕴涵代数和MV代数等代数结构,而且从中更能清楚地看出它们之间的密切联系,也将有助于对相应形式逻辑系统与模糊推理的研究．     

Spreads in Strongly Regular Graphs

A spread of a strongly regular graph is a partitionof the vertex set into cliques that meet Delsarte's bound (alsocalled Hoffman's bound). Such spreads give rise to coloringsmeeting Hoffman's lower bound for the chromatic number and tocertain imprimitive three-class association schemes. These correspondenceslead to conditions for existence. Most examples come from spreadsand fans in (partial) geometries. We give other examples, includinga spread in the McLaughlin graph. For strongly regular graphsrelated to regular two-graphs, spreads give lower bounds forthe number of non-isomorphic strongly regular graphs in the switchingclass of the regular two-graph.     

Partial ovoids and partial spreads in hermitian polar spaces

We present improved lower bounds on the sizes of small maximal partial ovoids in the classical hermitian polar spaces, and improved upper bounds on the sizes of large maximal partial spreads in the classical hermitian polar spaces. Of particular importance is the presented upper bound on the size of a maximal partial spread of H(3,q ²). For q = 2,3, the presented upper bound is sharp. For q = 3, our results confirm via theoretical arguments properties, deduced by computer searches performed by Ebert and Hirschfeld, for the largest partial spreads of H(3,9). An overview of the status regarding these results is given in two summarizing tables. The similar results for the classical symplectic and orthogonal polar spaces are presented in De Beule et al. [8].