On the connectedness of the complement of a ball in distance-regular graphs

An important property of strongly regular graphs is that the second subconstituent of any primitive strongly regular graph is always connected. Brouwer asked to what extent this statement can be generalized to distance-regular graphs. In this paper, we show that if γ is any vertex of a distance-regular graph Γ and t is the index where the standard sequence corresponding to the second largest eigenvalue of Γ changes sign, then the subgraph induced by the vertices at distance at least t from γ, is connected.     

The spectral excess theorem for distance-regular graphs having distance-<Emphasis Type="Italic">d</Emphasis> graph with fewer distinct eigenvalues

Let \(\Gamma \) be a distance-regular graph with diameter d and Kneser graph \(K=\Gamma _d\), the distance-d graph of \(\Gamma \). We say that \(\Gamma \) is partially antipodal when K has fewer distinct eigenvalues than \(\Gamma \). In particular, this is the case of antipodal distance-regular graphs (K with only two distinct eigenvalues) and the so-called half-antipodal distance-regular graphs (K with only one negative eigenvalue). We provide a characterization of partially antipodal distance-regular graphs (among regular graphs with \(d+1\) distinct eigenvalues) in terms of the spectrum and the mean number of vertices at maximal distance d from every vertex. This can be seen as a more general version of the so-called spectral excess theorem, which allows us to characterize those distance-regular graphs which are half-antipodal, antipodal, bipartite, or with Kneser graph being strongly regular.     

Geometric aspects of 2-walk-regular graphs

A t-walk-regular graph is a graph for which the number of walks of given length between two vertices depends only on the distance between these two vertices, as long as this distance is at most t. Such graphs generalize distance-regular graphs and t-arc-transitive graphs. In this paper, we will focus on 1- and in particular 2-walk-regular graphs, and study analogues of certain results that are important for distance-regular graphs. We will generalize Delsarte?s clique bound to 1-walk-regular graphs, Godsil?s multiplicity bound and Terwilliger?s analysis of the local structure to 2-walk-regular graphs. We will show that 2-walk-regular graphs have a much richer combinatorial structure than 1-walk-regular graphs, for example by proving that there are finitely many non-geometric 2-walk-regular graphs with given smallest eigenvalue and given diameter (a geometric graph is the point graph of a special partial linear space); a result that is analogous to a result on distance-regular graphs. Such a result does not hold for 1-walk-regular graphs, as our construction methods will show.     

Completely Regular Designs of Strength One

We study a class of highly regular t-designs. These are the subsets of vertices of the Johnson graph which are completely regular in the sense of Delsarte [2]. In [9], Meyerowitz classified the completely regular designs having strength zero. In this paper, we determine the completely regular designs having strength one and minimum distance at least two. The approach taken here utilizes the incidence matrix of (t+1)-sets versus k-sets and is related to the representation theory of distance-regular graphs [1, 5].     

On bipartite distance-regular graphs with a strongly closed subgraph of diameter three

Let Γ denote a distance-regular graph with a strongly closed regular subgraph Y. Hosoya and Suzuki [R. Hosoya, H. Suzuki, Tight distance-regular graphs with respect to subsets, European J. Combin. 28 (2007) 61-74] showed an inequality for the second largest and least eigenvalues of Γ in the case Y is of diameter 2. In this paper, we study the case when Γ is bipartite and Y is of diameter 3, and obtain an inequality for the second largest eigenvalue of Γ. Moreover, we characterize the distance-regular graphs with a completely regular strongly closed subgraph H(3,2).     

A simple proof of the spectral excess theorem for distance-regular graphs

The spectral excess theorem provides a quasi-spectral characterization for a (regular) graph Γ with d+1 distinct eigenvalues to be distance-regular graph, in terms of the excess (number of vertices at distance d) of each of its vertices. The original approach, due to Fiol and Garriga in 1997, was obtained by using a local approach, so giving a characterization of the so-called pseudo-distance-regularity around a vertex. In this paper we present a new simple projection method based in a global point of view, and where the mean excess plays an essential role.     

The geometry of t‐spreads in k‐walk‐regular graphs

A graph is walk‐regular if the number of closed walks of length ? rooted at a given vertex is a constant through all the vertices for all ?. For a walk‐regular graph G with d+1 different eigenvalues and spectrally maximum diameter D=d, we study the geometry of its d‐spreads, that is, the sets of vertices which are mutually at distance d. When these vertices are projected onto an eigenspace of its adjacency matrix, we show that they form a simplex (or tetrahedron in a three‐dimensional case) and we compute its parameters. Moreover, the results are generalized to the case of k‐walk‐regular graphs, a family which includes both walk‐regular and distance‐regular graphs, and their t‐spreads or vertices at distance t from each other. © 2009 Wiley Periodicals, Inc. J Graph Theory 64:312–322, 2010     

Outerplanar and Planar Oriented Cliques

The clique number of an undirected graph G is the maximum order of a complete subgraph of G and is a well‐known lower bound for the chromatic number of G. Every proper k‐coloring of G may be viewed as a homomorphism (an edge‐preserving vertex mapping) of G to the complete graph of order k. By considering homomorphisms of oriented graphs (digraphs without cycles of length at most 2), we get a natural notion of (oriented) colorings and oriented chromatic number of oriented graphs. An oriented clique is then an oriented graph whose number of vertices and oriented chromatic number coincide. However, the structure of oriented cliques is much less understood than in the undirected case. In this article, we study the structure of outerplanar and planar oriented cliques. We first provide a list of 11 graphs and prove that an outerplanar graph can be oriented as an oriented clique if and only if it contains one of these graphs as a spanning subgraph. Klostermeyer and MacGillivray conjectured that the order of a planar oriented clique is at most 15, which was later proved by Sen. We show that any planar oriented clique on 15 vertices must contain a particular oriented graph as a spanning subgraph, thus reproving the above conjecture. We also provide tight upper bounds for the order of planar oriented cliques of girth k for all .     

The impact of 2-neighborhoods on graph structure

Connected regular graphs of diameter greater than 2 with 2-neighborhoods isomorphic to a certain graph Δ of diameter 2 are studied. Regular graphs of diameter 2 with 2-neighborhoods isomorphic to the distance-regular graph Δ of diameter 3 are also studied. It is prowed that 2-locally Schrikhande graphs do not exist. Translated fromMatematicheskie Zametki, Vol. 62, No. 6, pp. 892–897, December, 1997. Translated by S. S. Anisov     

On almost distance-regular graphs

Distance-regular graphs are a key concept in Algebraic Combinatorics and have given rise to several generalizations, such as association schemes. Motivated by spectral and other algebraic characterizations of distance-regular graphs, we study ‘almost distance-regular graphs’. We use this name informally for graphs that share some regularity properties that are related to distance in the graph. For example, a known characterization of a distance-regular graph is the invariance of the number of walks of given length between vertices at a given distance, while a graph is called walk-regular if the number of closed walks of given length rooted at any given vertex is a constant. One of the concepts studied here is a generalization of both distance-regularity and walk-regularity called m-walk-regularity. Another studied concept is that of m-partial distance-regularity or, informally, distance-regularity up to distance m. Using eigenvalues of graphs and the predistance polynomials, we discuss and relate these and other concepts of almost distance-regularity, such as their common generalization of (?,m)-walk-regularity. We introduce the concepts of punctual distance-regularity and punctual walk-regularity as a fundament upon which almost distance-regular graphs are built. We provide examples that are mostly taken from the Foster census, a collection of symmetric cubic graphs. Two problems are posed that are related to the question of when almost distance-regular becomes whole distance-regular. We also give several characterizations of punctually distance-regular graphs that are generalizations of the spectral excess theorem.     

Distance-residual subgraphs

For a connected finite graph G and a subset V₀ of its vertex set, a distance-residual subgraph is a subgraph induced on the set of vertices at the maximal distance from V₀. Some properties and examples of distance-residual subgraphs of vertex-transitive, edge-transitive, bipartite and semisymmetric graphs are presented. The relations between the distance-residual subgraphs of product graphs and their factors are explored.     

Distance-regular graphs of large diameter that are completely regular clique graphs

A connected graph is said to be a completely regular clique graph with parameters (s, c), \(s, c \in {\mathbb {N}}\), if there is a collection \(\mathcal {C}\) of completely regular cliques of size \(s+1\) such that every edge is contained in exactly c members of \(\mathcal {C}\). It is known that many families of distance-regular graphs are completely regular clique graphs. In this paper, we determine completely regular clique graph structures, i.e., the choices of \(\mathcal {C}\), of all known families of distance-regular graphs with unbounded diameter. In particular, we show that all distance-regular graphs in this category are completely regular clique graphs except the Doob graphs, the twisted Grassmann graphs and the Hermitean forms graphs. We also determine parameters (s, c); however, in a few cases we determine only s and give a bound on the value c. Our result is a generalization of a series of works by J. Hemmeter and others who determined distance-regular graphs in this category that are bipartite halves of bipartite distance-regular graphs.     

Arc-transitive distance-regular covers of cliques with <Emphasis Type="Italic">λ</Emphasis> = µ

We study antipodal distance-regular graphs of diameter 3 such that their automorphism group acts transitively on the set of pairs (a, b), where {a, b} is an edge of the graph. Since the automorphism group of such graphs acts 2-transitively on the set of antipodal classes, the classification of 2-transitive permutation groups can be used. We classify arc-transitive distance-regular graphs of diameter 3 in which any two vertices at distance at most two have exactly µ common neighbors.     

Game-perfect graphs

A graph coloring game introduced by Bodlaender (Int J Found Comput Sci 2:133–147, 1991) as coloring construction game is the following. Two players, Alice and Bob, alternately color vertices of a given graph G with a color from a given color set C, so that adjacent vertices receive distinct colors. Alice has the first move. The game ends if no move is possible any more. Alice wins if every vertex of G is colored at the end, otherwise Bob wins. We consider two variants of Bodlaender’s graph coloring game: one (A) in which Alice has the right to have the first move and to miss a turn, the other (B) in which Bob has these rights. These games define the A-game chromatic number resp. the B-game chromatic number of a graph. For such a variant g, a graph G is g-perfect if, for every induced subgraph H of G, the clique number of H equals the g-game chromatic number of H. We determine those graphs for which the game chromatic numbers are 2 and prove that the triangle-free B-perfect graphs are exactly the forests of stars, and the triangle-free A-perfect graphs are exactly the graphs each component of which is a complete bipartite graph or a complete bipartite graph minus one edge or a singleton. From these results we may easily derive the set of triangle-free game-perfect graphs with respect to Bodlaender’s original game. We also determine the B-perfect graphs with clique number 3. As a general result we prove that complements of bipartite graphs are A-perfect.      

On simplices in diameter graphs in ℝ4

Agraph G is a diameter graph in ?^d if its vertex set is a finite subset in ?^d of diameter 1 and edges join pairs of vertices a unit distance apart. It is shown that if a diameter graph G in ?⁴ contains the complete subgraph K on five vertices, then any triangle in G shares a vertex with K. The geometric interpretation of this statement is as follows. Given any regular unit simplex on five vertices and any regular unit triangle in ?⁴, then either the simplex and the triangle have a common vertex or the diameter of the union of their vertex sets is strictly greater than 1.

     

More on block intersection polynomials and new applications to graphs and block designs

The concept of intersection numbers of order r for t-designs is generalized to graphs and to block designs which are not necessarily t-designs. These intersection numbers satisfy certain integer linear equations involving binomial coefficients, and information on the non-negative integer solutions to these equations can be obtained using the block intersection polynomials introduced by P.J. Cameron and the present author. The theory of block intersection polynomials is extended, and new applications of these polynomials to the studies of graphs and block designs are obtained. In particular, we obtain a new method of bounding the size of a clique in an edge-regular graph with given parameters, which can improve on the Hoffman bound when applicable, and a new method for studying the possibility of a graph with given vertex-degree sequence being an induced subgraph of a strongly regular graph with given parameters.     

Clique trees of chordal graphs: leafage and 3-asteroidals

Chordal graphs were characterized as those graphs having a tree, called clique tree, whose vertices are the cliques of the graph and for every vertex in the graph, the set of cliques that contain it form a subtree of clique tree. In this work, we study the relationship between the clique trees of a chordal graph and its subgraphs. We will prove that clique trees can be described locally and all clique trees of a graph can be obtained from clique trees of subgraphs. In particular, we study the leafage of chordal graphs, that is the minimum number of leaves among the clique trees of the graph. It is known that interval graphs are chordal graphs without 3-asteroidals. We will prove a generalization of this result using the framework developed in the present article. We prove that in a clique tree that realizes the leafage, for every vertex of degree at least 3, and every choice of 3 branches incident to it, there is a 3−asteroidal in these branches.     

A Characterization of the Hamming Graphs and the Dual Polar Graphs by Completely Regular Subgraphs

In this paper we study a distance-regular graph Γ of diameter d ≥? 3 which satisfies the following two conditions: (a) For any integer i with 1 ≤? i ≤? d ? 1 and for any pair of vertices at distance i in Γ there exists a strongly closed subgraph of diameter i containing them; (b) There exists a strongly closed subgraph Δ which is completely regular in Γ. It is known that if Δ has diameter 1, then Γ is a regular near polygon. We prove that if a strongly closed subgraph Δ of diameter j with 2 ≤? j ≤? d ? 1 is completely regular of covering radius d ? j in Γ, then Γ is either a Hamming graph or a dual polar graph.     

Automorphisms of coverings of strongly regular graphs with parameters (81,20,1,6)

The possible orders and subgraphs of fixed points of the automorphisms of distance-regular graphs of diameter 4 which are r-coverings of a strongly regular graph with parameters (81, 20, 1, 6) for r ∈ {2, 3, 6} are found. It is proved that, if the automorphism group of a covering of the above type acts transitively on the set of vertices of the graph, then r = 3.     

Distance-Regular Graphs with Strongly Regular Subconstituents

In [3] Cameron et al. classified strongly regular graphs with strongly regular subconstituents. Here we prove a theorem which implies that distance-regular graphs with strongly regular subconstituents are precisely the Taylor graphs and graphs with a ₁ = 0 and a _i {0,1} for i = 2,...,d.