Feasibility conditions for the existence of walk-regular graphs

A graph X is walk-regular if the vertex-deleted subgraphs of X all have the same characteristic polynomial. Examples of such graphs are vertex-transitive graphs and distance-regular graphs. We show that the usual feasibility conditions for the existence of a distance-regular graph with a given intersection array can be extended so that they apply to walk-regular graphs. Despite the greater generality, our proofs are more elementary than those usually given for distance-regular graphs. An application to the computation of vertex-transitive graphs is described.     

Triangle-free strongly circular-perfect graphs

Zhu [X. Zhu, Circular-perfect graphs, J. Graph Theory 48 (2005) 186-209] introduced circular-perfect graphs as a superclass of the well-known perfect graphs and as an important χ-bound class of graphs with the smallest non-trivial χ-binding function χ(G)≤ω(G)+1. Perfect graphs have been recently characterized as those graphs without odd holes and odd antiholes as induced subgraphs [M. Chudnovsky, N. Robertson, P. Seymour, R. Thomas, The strong perfect graph theorem, Ann. Math. (in press)]; in particular, perfect graphs are closed under complementation [L. Lovász, Normal hypergraphs and the weak perfect graph conjecture, Discrete Math. 2 (1972) 253-267]. To the contrary, circular-perfect graphs are not closed under complementation and the list of forbidden subgraphs is unknown.We study strongly circular-perfect graphs: a circular-perfect graph is strongly circular-perfect if its complement is circular-perfect as well. This subclass entails perfect graphs, odd holes, and odd antiholes. As the main result, we fully characterize the triangle-free strongly circular-perfect graphs, and prove that, for this graph class, both the stable set problem and the recognition problem can be solved in polynomial time.Moreover, we address the characterization of strongly circular-perfect graphs by means of forbidden subgraphs. Results from [A. Pêcher, A. Wagler, On classes of minimal circular-imperfect graphs, Discrete Math. (in press)] suggest that formulating a corresponding conjecture for circular-perfect graphs is difficult; it is even unknown which triangle-free graphs are minimal circular-imperfect. We present the complete list of all triangle-free minimal not strongly circular-perfect graphs.     

Disconnecting strongly regular graphs

In this paper, we show that the minimum number of vertices whose removal disconnects a connected strongly regular graph into non-singleton components equals the size of the neighborhood of an edge for many graphs. These include block graphs of Steiner 2-designs, many Latin square graphs and strongly regular graphs whose intersection parameters are at most a quarter of their valency.     

Directed triangles in directed graphs

We show that each directed graph on n vertices, each with indegree and outdegree at least n/t, where

Full-size image

, contains a directed circuit of length at most 3.     

Directed triangles in directed graphs

We show that each directed graph (with no parallel arcs) on n vertices, each with indegree and outdegree at least n/twhere t=2.888997… contains a directed circuit of length at most 3.     

Meyniel graphs are strongly perfect

Meyniel (Discrete Math.16 (1976), 339–342) proved that a graph is perfect whenever each of its odd cycles of length at least five has at least two chords. This result is strengthened by proving that every graph satisfying Meyniel's condition is strongly perfect (i.e., each of its induced subgraphs H contains a stable set which meets all the maximal cliques in H).     

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.     

Some new strongly regular graphs

We show that three pairwise 4-regular graphs constructed by the second author are members of infinite families.     

5-chromatic strongly regular graphs

In this paper, we begin the determination of all primitive strongly regular graphs with chromatic number equal to 5. Using eigenvalue techniques, we show that there are at most 43 possible parameter sets for such a graph. For each parameter set, we must decide which strongly regular graphs, if any, possessing the set are 5-chromatic. In this way, we deal completely with 34 of these parameter sets using eigenvalue techniques and computer enumerations.     

New families of strongly regular graphs

We apply symmetric balanced generalized weighing matrices with zero diagonal to construct four parametrically new infinite families of strongly regular graphs. © 2003 Wiley Periodicals, Inc. J Combin Designs 11: 208–217, 2003; Published online in Wiley InterScience ( www.interscience.wiley.com ). DOI 10.1002/jcd.10038     

A decomposition for strongly perfect graphs

A graph G is strongly perfect if every induced subgraph H of G contains a stable set that meets all the maximal cliques of H. We present a graph decomposition that preserves strong perfection: more precisely, a stitch decomposition of a graph G = (V, E) is a partition of V into nonempty disjoint subsets V₁, V₂ such that in every P₄ with vertices in both V_iapos;s, each of the three edges has an endpoint in V₁ and the other in V₂. We give a good characterization of graphs that admit a stitch decomposition and establish several results concerning the stitch decomposition of strongly perfect graphs.     

Unimodular lattices and strongly regular graphs

One finds a relationship between the theory of unimodular lattices and the theory of strongly regular graphs. For a unimodular, even lattice of dimension 32 having a system of roots of type A₁ one constructs a strongly regular graph with parameters n=8184, a=7595, c=7042, d=7130. The graphs that arise from certain Steiner sixtuple systems have the same parameters. One also constructs strongly regular graphs for extremal lattices of dimension 48.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 129, pp. 30–38, 1983.     

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     

Chordal bipartite, strongly chordal, and strongly chordal bipartite graphs

Robert E. Jamison characterized chordal graphs by the edge set of every k-cycle being the symmetric difference of k−2 triangles. Strongly chordal (and chordal bipartite) graphs can be similarly characterized in terms of the distribution of triangles (respectively, quadrilaterals). These results motivate a definition of ‘strongly chordal bipartite graphs’, forming a class intermediate between bipartite interval graphs and chordal bipartite graphs.     

Directed graphs of inner translations of semigroups

A mapping \(\alpha :S\rightarrow S\) is called a Cayley function if there exist an associative operation \(\mu :S\times S\rightarrow S\) and an element \(a\in S\) such that \(\alpha (x)=\mu (a,x)\) for every \(x\in S\). The aim of the paper is to give a characterization of Cayley functions in terms of their directed graphs. This characterization is used to determine which elements of the centralizer of a permutation on a finite set are Cayley functions. The paper ends with a number of problems.