Some general constructions of geodetic blocks

A general construction procedure for geodetic blocks, starting from an arbitrary geodetic block is given, which unifies and generalizes many of the known general procedures. The construction of geodetic blocks homeomorphic to a given one is also analysed and the problem of Bosák on the existence of such a graph for the Hoffman-Singleton graph is settled in the affirmative. A simple characterization theorem for geodetic blocks is given and the existence of geodetic blocks with given girth and diameter is investigated.     

Subgroup-Chain Graphs

Starting from chains of subgroups in a group a construction for k-regular graphs is given and as an application of this construction a cubic graph of girth 20 with 6072 vertices is produced. This graph is exceptional in that, at the time of writing, it has fewest vertices for a cubic graph of this girth.     

图G为自构线图的充要条件

首次给出自构线图的定义,并证明:简单图G为自构线图的充要条件是图G为2-正则简单图.     

Every finite graph is a full subgraph of a rigid graph

We prove that every finite undirected graph is a full subgraph of a rigid graph. Our construction proceeds on taking a family of “mutually rigid” graphs and attaching them to the vertices of a given graph in a one-to-one manner; then the vertices are fixed on their place. Actually, the new graph is “strongly rigid”, which enables us to show that the category of all graphs containing a given finite graph as a full subgraph is binding.     

Maximal and extremal singular graphs

A graph G is singular if the nullspace of its adjacency matrix is nontrivial. Such a graph contains induced subgraphs called singular configurations of nullity 1. We present two algorithms. One is for the construction of a maximal singular nontrivial graph G containing an induced subgraph, which is a singular configuration with the support of a vector in its nullspace as in that of G. The second is for the construction of a nut graph, a graph of nullity one whose null vector has no zero entries. An extremal singular graph of a given order, with the maximal nullity and support, has a nut graph as a maximal singular configuration.     

图的线独立数的一个公式

本文中我们用等秩变换证明了连通图G的所有生成树的邻接矩阵的秩中最大者就是图G线独立数的两倍。特别,我们给出了连通图G具有完美匹配的一个新的充要条件。     

Locally-finite connected-homogeneous digraphs

A digraph is connected-homogeneous if any isomorphism between finite connected induced subdigraphs extends to an automorphism of the digraph. We consider locally-finite connected-homogeneous digraphs with more than one end. In the case that the digraph embeds a triangle we give a complete classification, obtaining a family of tree-like graphs constructed by gluing together directed triangles. In the triangle-free case we show that these digraphs are highly arc-transitive. We give a classification in the two-ended case, showing that all examples arise from a simple construction given by gluing along a directed line copies of some fixed finite directed complete bipartite graph. When the digraph has infinitely many ends we show that the descendants of a vertex form a tree, and the reachability graph (which is one of the basic building blocks of the digraph) is one of: an even cycle, a complete bipartite graph, the complement of a perfect matching, or an infinite semiregular tree. We give examples showing that each of these possibilities is realised as the reachability graph of some connected-homogeneous digraph, and in the process we obtain a new family of highly arc-transitive digraphs without property Z.     

Embedding arbitrary finite simple graphs into small strongly regular graphs

It is well known that any finite simple graph Γ is an induced subgraph of some exponentially larger strongly regular graph Γ (e.g., [2, 8]). No general polynomial‐size construction has been known. For a given finite simple graph Γ on υ vertices, we present a construction of a strongly regular graph Γ on O(υ⁴) vertices that contains Γ as its induced subgraph. A discussion is included of the size of the smallest possible strongly regular graph with this property. © 2000 John Wiley & Sons, Inc. J Graph Theory 34: 1–8, 2000     

Tutte uniqueness of line graphs

We prove that if a graph H has the same Tutte polynomial as the line graph of a d-regular, d-edge-connected graph, then H is the line graph of a d-regular graph. Using this result, we prove that the line graph of a regular complete t-partite graph is uniquely determined by its Tutte polynomial. We prove the same result for the line graph of any complete bipartite graph.     

Random graphs with given vertex degrees and switchings

Random graphs with a given degree sequence are often constructed using the configuration model, which yields a random multigraph. We may adjust this multigraph by a sequence of switchings, eventually yielding a simple graph. We show that, assuming essentially a bounded second moment of the degree distribution, this construction with the simplest types of switchings yields a simple random graph with an almost uniform distribution, in the sense that the total variation distance is o(1). This construction can be used to transfer results on distributional convergence from the configuration model multigraph to the uniform random simple graph with the given vertex degrees. As examples, we give a few applications to asymptotic normality. We show also a weaker result yielding contiguity when the maximum degree is too large for the main theorem to hold.     

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.     

Fractional chromatic numbers of cones over graphs

We introduce a construction called the cone over a graph. It is a natural generalisation of Mycielski's construction. We give a formula for the fractional chromatic numbers of all cones over graphs, which generalizes that given in 3 for Mycielski's construction. © 2001 John Wiley & Sons, Inc. J Graph Theory 38: 87–94, 2001     

On natural isomorphisms of cycle permutation graphs

Two problems are approached in this paper. Given a permutation onn elements, which permutations onn elements yield cycle permutation graphs isomorphic to the cycle permutation graph yielded by the given permutation? And, given two cycle permutation graphs, are they isomorphic? Here the author deals only with natural isomorphisms, those isomorphisms which map the outer and inner cycles of one cycle permutation graph to the outer and inner cycles of another cycle permutation graph. A theorem is stated which then allows the construction of the set of permutations which yield cycle permutation graphs isomorphic to a given cycle permutation graph by a natural isomorphism. Another theorem is presented which finds the number of such permutations through the use of groups of symmetry of certain drawings of cycles in the plane. These drawings are also used to determine whether two given cycle permutation graphs are isomorphic by a natural isomorphism. These two methods are then illustrated by using them to solve the first problem, restricted to natural isomorphism, for a certain class of cycle permutation graphs.     

Characterizations of second iterated line graphs

The main results of this paper are two characterizations of second iterated line graphs, i.e., two sets of necessary and sufficient conditions for the existence of solutions to the graph equation H = L²(G). A method to get a root of the graph equation is also given if one exists.     

The square of every two-connected graph is Hamiltonian

A graph G is a line-critical block if κ(G) = 2 and if for any line e of G the graph G ? e has κ(G ? e) = 1.If G is a line-critical block, then G is either a DT-block (i.e., G is a two-connected graph in which every line is incident to a point of degree two), or G contains a specific two-connected subgraph which is a DT-block (Theorem 1). Using this result and results of the preceding paper on DT-graphs, a simple proof of the conjecture that the square of every two-connected graph is Hamiltonian is given.     

A spectral characterization of the line graph of a BIBD with λ=1

The line graph of a balanced incomplete block design has as its vertices the ordered pairs (v, B), where B is a block and v is a variety contained in that block. Two vertices are adjacent if they agree in one coordinate Since the line graphs are isomorphic only when the designs are isomorphic, various characterizations of the line graph of a BIBD have been given. The following is proved herein: A BIBD with λ=1 is characterized by its spectrum if r+k>18.     

Constructing universal graphs for induced-hereditary graph properties

Rado constructed a (simple) denumerable graph R with the positive integers as vertex set with the following edges: For given m and n with m < n, m is adjacent to n if n has a 1 in the m’th position of its binary expansion. It is well known that R is a universal graph in the set I of all countable graphs (since every graph in I is isomorphic to an induced subgraph of R). In this paper we describe a general recursive construction which proves the existence of a countable universal graph for any induced-hereditary property of countable general graphs. A general construction of a universal graph for the set of finite graphs in a product of properties of graphs is also presented. The paper is concluded by a comparison between the two types of construction of universal graphs.     

Four Constructions of Highly Symmetric Tetravalent Graphs

Given a connected, dart‐transitive, cubic graph, constructions of its Hexagonal Capping and its Dart Graph are considered. In each case, the result is a tetravalent graph which inherits symmetry from the original graph and is a covering of the line graph.Similar constructions are then applied to a map (a cellular embedding of a graph in a surface) giving tetravalent coverings of the medial graph. For each construction, conditions on the graph or the map to make the constructed graph dart‐transitive, semisymmetric or ‐transitive are considered.     

Automorphic graphs and the Krein condition

An automorphic graph is a distance-transitive graph, not a complete graph or a line graph, whose automorphism group acts primitively on the vertices. This paper shows that, for small values of the valency and diameter, such graphs are rare. The basic tool is the intersection array, for which there are several very restrictive feasibility conditions. In particular, a slight generalisation of the Krein condition of Scott and Higman is given, with a simplified proof.