Divisible design graphs from symplectic graphs over rings with a unique non-trivial ideal

The symplectic graph $Sp(2d, q)$ is the collinearity graph of the symplectic space of dimension $2d$ over the finite field of order $q$. A $k$-regular graph on $v$ vertices is a divisible design graph with parameters $(v,k,\lambda_1,\lambda_2,m,n)$ if its vertex set can be partitioned into $m$ classes of size $n$, such that any two different vertices from the same class have $\lambda_1$ common neighbours, and any two vertices from different classes have $\lambda_2$ common neighbours whenever it is not complete or edgeless. In this paper we propose a new construction of strongly regular graphs with the parameters of the complement of the symplectic graph using divisible design graphs.

Divisible design graphs from the symplectic graph

Divisible design graphs

Divisible design graphs were introduced in 2011 by Haemers, Kharaghani and Meulenberg. Further, divisible design graphs which can be obtained as Cayley graphs were recently studied by Kabanov and Shalaginov. In this paper we give new constructions of divisible design Cayley graphs and classify divisible design Cayley graphs on \(v \le 27\) vertices.

A divisible design graph is a graph whose adjacency matrix is the incidence matrix of a divisible design. These graphs are a natural generalization of (v, k, ⋋)-graphs. In this paper we develop some theory, find many parameter conditions and give several constructions.

A divisible design graph (DDG for short) is a graph whose adjacency matrix is the incidence matrix of a divisible design. DDGs were introduced by Kharaghani, Meulenberg and the second author as a generalization of $$(v,k,\lambda )$$ ( v , k , ? ) -graphs. It turns out that most (but not all) of the known examples of DDGs are walk-regular. In this paper we present an easy criterion for this to happen. In several cases walk-regularity is forced by the parameters of the DDG; then known conditions for walk-regularity lead to nonexistence results for DDGs. In addition, we construct some new DDGs, and check old and new constructions for walk-regularity. In doing so, we present and use special properties in case the classes have size two. All feasible parameter sets for DDGs on at most $$27$$ 27 vertices are examined. Existence is established in all but one case, and existence of a walk-regular DDG in all cases.

Self-orthogonal codes constructed from orbit matrices of 2-designs and quotient matrices of divisible designs

A $k$-regular graph is called a divisible design graph if its vertex set can be partitioned into $m$ classes of size $n$, such that two distinct vertices from the same class have exactly $\lambda_1$ common neighbours, and two vertices from different classes have exactly $\lambda_2$ common neighbours. In this paper, we find the vertex connectivity of some classes of divisible design graphs, in particular, we present examples of divisible design graphs, whose vertex connectivity is less than $k$, where $k$ is the degree of a vertex. We also show that the vertex connectivity a divisible design graphs may be less than $k$ by any power of 2.

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.

New versions of the Wallis-Fon-Der-Flaass construction to create divisible design graphs

q-Analogs of divisible design graphs and Deza graphs

A Deza graph with parameters is a k-regular graph with n vertices such that any two of its vertices have b or a common neighbours, where . In this paper we investigate spectra of Deza graphs. In particular, using the eigenvalues of a Deza graph we determine the eigenvalues of its children. Divisible design graphs are significant cases of Deza graphs. Sufficient conditions for Deza graphs to be divisible design graphs are given, a few families of divisible design graphs are presented and their properties are studied. Our special attention goes to the invertibility of the adjacency matrices of Deza graphs.

Strongly regular graphs decomposable into a divisible design graph and a Delsarte clique

We present a construction that gives an infinite series of divisible design\ngraphs which are Cayley graphs.\n

Abstract: Divisible design graphs (DDG for short) have been recently defined by Kharaghani, Meulenberg and the second author as a generalization of (v, k, λ)-graphs. In this paper we give some new constructions of DDGs, most of them using Hadamard matrices and (v, k, λ)-graphs. For three parameter sets we give a nonexistence proof. Furthermore, we find conditions for a DDG to be walk-regular. It follows that most of the known examples are walk-regular, but some are not. In case walk-regularity of a DDG is forced by the parameters, necessary conditions for walk-regularity lead to new nonexistence results for DDGs. We examine all feasible parameter sets for DDGs on at most 27 vertices, establish existence in all but one cases, and decide on existence of a walk-regular DDG in all cases.