Approximation Schemes for the Generalized Traveling Salesman Problem

A graph Γ is called a Deza graph if it is regular and the number of common neighbors of any two distinct vertices is one of two fixed values. A Deza graph is called a strictly Deza graph if it has diameter 2 and is not strongly regular. In 1992, Gardiner et al. proved that a strongly regular graph that contains a vertex with disconnected second neighborhood is a complete multipartite graph with parts of the same size greater than 2. In this paper, we study strictly Deza graphs with disconnected second neighborhoods of vertices. In Section 2, we prove that, if each vertex of a strictly Deza graph has disconnected second neighborhood, then the graph is either edge-regular or coedge-regular. In Sections 3 and 4, we consider strictly Deza graphs that contain at least one vertex with disconnected second neighborhood. In Section 3, we show that, if such a graph is edge-regular, then it is the s-coclique extension of a strongly regular graph with parameters (n, k, λ, μ), where s is an integer, s ≥ 2, and λ = μ. In Section 4, we show that, if such a graph is coedge-regular, then it is the 2-clique extension of a complete multipartite graph with parts of the same size greater than or equal to 3.     

On WL-rank of Deza Cayley graphs

The WL-rank of a graph Γ is defined to be the rank of the coherent configuration of Γ. We construct a new infinite family of strictly Deza Cayley graphs for which the WL-rank is equal to the number of vertices. The graphs from this family are divisible design and integral.     

On <Emphasis Type="Italic">K</Emphasis><Subscript>1,3</Subscript>-free strictly Deza graphs

A graph on v vertices is called a Deza graph with parameters (v, k, b, a) if it is k-regular and the number of common neighbors of two distinct vertices takes on one of two values. We describe strictly Deza graphs that do not contain K _1,3 among their induced subgraphs and are unions of closed neighborhoods of two nonadjacent vertices. The latter condition means that there are two nonadjacent vertices such that any other vertex is adjacent to at least one of the them.     

Strictly Deza line graphs

For a given graph G, its line graph L(G) is defined as the graph with vertex set equal to the edge set of G in which two vertices are adjacent if and only if the corresponding edges of G have exactly one common vertex. A k-regular graph of diameter 2 on υ vertices is called a strictly Deza graph with parameters (υ, k, b, a) if it is not strongly regular and any two vertices have a or b common neighbors. We give a classification of strictly Deza line graphs.     

A control problem with dominating uncertainty

We find the vertex connectivity number of Deza graphs obtained from strongly regular graphs with the use of an automorphism of order 2.     

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.     

Spectra of strongly Deza graphs

A Deza graph G with parameters $(n,k,b,a)$ is a k-regular graph with n vertices such that any two distinct vertices have b or a common neighbours. The children $G_A$ and $G_B$ of a Deza graph G are defined on the vertex set of G such that every two distinct vertices are adjacent in $G_A$ or $G_B$ if and only if they have a or b common neighbours, respectively. A strongly Deza graph is a Deza graph with strongly regular children. In this paper we give a spectral characterisation of strongly Deza graphs, show relationships between eigenvalues, and study strongly Deza graphs which are distance-regular.     

Singular linear space and its applications

As a generalization of attenuated spaces, the concept of singular linear spaces was introduced in [K. Wang, J. Guo, F. Li, Association schemes based on attenuated spaces, European J. Combin. 31 (2010) 297–305]. This paper first gives two anzahl theorems in singular linear spaces, and then discusses their applications to the constructions of Deza digraphs, quasi-strongly regular graphs, lattices and authentication codes.     

Approximation in area of graphs with isolated singularities

It is shown that graphs with finite mass of mappings which are smooth except for isolated points, where they are continuous, can be approximated in area by graphs of smooth maps. The continuity hypothesis is then removed if the dimension n is strictly larger than the codimension N or if the dimension n is at least 3.