It is a distance transitive graph see the foster census and therefore distance regular there are 24 perfect matchings in the heawood graph. Several other important regular combinatorial structures are then shown to be equivalent to special families of distance regular graphs. Lambeck department of mathematics and computing science, eindhoven university of technology, p. A strongly regular graph is a distance regular graph with diameter 2 whenever. Distance regular graphs or a newer source, which could be better for beginners in this area. On the multiplicity of eigenvalues of distance regular graphs the main goal of this note is to give better lower bounds on the multiplicity of eigenvalues in a distance regular graph. Every smaller cubic graph has shorter cycles, so this graph is the 6cage, the smallest cubic graph of girth 6. Improving diameter bounds for distanceregular graphs. Merging the first and third classes in a connected graph is the operation of adding edges between all vertices at distance 3 in the original graph while keeping the original edges. The source is some troff dialect, with most formatting commands removed. Algebraic graph theory is the branch of mathematics that studies graphs by using.
Implementing brouwers database of strongly regular graphs. The primitive distance transitive representations of the fischer groups linton, stephen a. Merging the first and third classes in bipartite distance. An interesting property of a class of circulant graphs. It is well known brouwer, cohen and neumaier 2, 11. A strongly distance regular graph is a distance regular graph g of diameter d, say with the property that its distance d graph g d is strongly regular. This solves a problem of brouwer, cohen, and neumaier 2, p.
Neumaier, distance regular graphs, springerverlag, 1989. Pasechnik 0 1 0 department of computer science, the university of oxford, oxford, uk 1 cnrs and universite parissud 11, orsay, france andries brouwer maintains a public database of existence results for strongly regular graphs on n. A reduction of the graph reconstruction conjecture in. Everyday low prices and free delivery on eligible orders. Some background material on distance regular graphs can be found here. New distance regular graphs arising from dimensional dual. Distanceregular graphs the electronic journal of combinatorics. A new distanceregular graph associated to the mathieu. Including the heawood graph, there are 8 distinct graphs of order 14 with crossing number 3. Folge a series of modern surveys in mathematics, vol 18.
A graph is said to be reconstructible if it is determined up to isomor phism from the collection of all its onevertex deleted unlabeled subgraphs. We introduce the concept of distance mean regular graph, which can be seen as a generalization of both vertextransitive and distance regular graphs. Starting from very elementary regularity properties, the concept of a distance regular graph arises naturally as a common setting for regular graphs which are extremal in one sense or another. Folge a series of modern surveys in mathematics on free shipping on qualified orders. We construct two families of distance regular graphs, namely the subgraph of the dual polar graph of type b 3q induced on the vertices far from a fixed point, and the subgraph of the dual polar graph of type d 4q induced on the vertices far from a fixed edge.
The reader is referred to brouwer cohen neumaier 5 and van damkoolentanaka 7 for more background information on distance regular graphs. Pdf parameter restrictions for distanceregular graphs. The graph is cubic, and all cycles in the graph have six or more edges. Graphs cospectral with h3, q which are disjoint unions of. This in uential monograph, which is almost like an encyclopedia of distanceregular graphs, inspired many researchers to work on distanceregular graphs, such as the authors of this survey. On the more special topic ofdistanceregular graphs a lot ofmaterial can be found in bicas 7. A search query can be a title of the book, a name of the author, isbn or anything else. From these results we prove the nonexistence of distanceregular graphs associated to 20 feasible intersection arrays from the book distanceregular graphs by brouwer, cohen and neumaier. Distance regularity of a graph may be checked in the grape package in gap using the function isdistanceregularg. Ever since the discovery of the five platonic solids in ancient times, the study of symmetry and regularity has been one of the most fascinating aspects of mathematics. On vertex decomposable and cohen macaulay regular graphs luviano, j. Strongly related to strongly regular graphs are regular twographs, and chapter 10 mainly discusses seidels work on sets of equiangular lines.
There exist four obvious families of such graphs, which are called trivial. Distance regular graphs which can be downloaded from the web. A connected graph g is distanceregular if for any nodes x,y and any integers i,j0,1. Conversely, if a distance regular graph has diameter two, then it is a strongly regular graph. Distance transitive graphs were first defined in 1971 by norman l. On distance regular graphs with cib1 sciencedirect. Two distanceregular graphs two distanceregular graphs brouwer, andries. The central problem in the theory of distance regular graphs is their classification, which seems to be very hard. Nonexistence of some antipodal distanceregular graphs of. Reliable information about the coronavirus covid19 is available from the world health organization current situation, international travel.
We construct 0, 2 graphs from root systems with simply laced diagram and study their properties. We prove that a distance regular graph with intersection array 56,36,9. Each strongly regular graph is a distance regular graph. A disconnected graph is distance regular iff it is a disjoint union of cospectral distance regular graphs. All necessary definitions tan be found in brouwer, most of them in l, section lar graphr, the vector cohen, and neumaier l. Dec 20, 2011 two distance regular graphs two distance regular graphs brouwer, andries. For more detailed information on distance regular graphs in general, and the graphs on this site, the following references are invaluable. We present an introduction to distanceregular graphs for the reader who is unfamiliar with the subject, and then give an overview of some developments in the area of distanceregular graphs since the monograph bcn brouwer, a. We present an introduction to distanceregular graphs for the reader who is. On distance regular graphs with connecting repositories. Distanceregular cayley graphs on dihedral groups, journal of.
Brouwer 93 constructed related distanceregular graphs with. By looking at the eigenvalues of distance regular graphs it is possible to get euclidean representa. Returns true if the graph is distance regular, false otherwise. In pyber showed that the diameter of distanceregular graphs is at most 5 times the 2 logarithm of the number of vertices.
It is known that the classical distance regular graphs contain many nice substructures, like designs in the johnson schemes and the binary golay code in the 23cuhe. Contributions to the theory of distance regular graphs. They are antipodal 2 2t\gamma1 fold covers of the complete graph on 2 2t vertices. The array of integers characterizing a distance regular graph is known as its intersection array. This interplay between regularity and symmetry properties of graphs is the theme of this book.
Distanceregular graphs with a1 or c2 at least half the valency. Numerous and frequentlyupdated resource results are available from this search. In pyber showed that the diameter of distance regular graphs is at most 5 times the 2 logarithm of the number of. In some cases, the constructed codes are also completely transitive and the corresponding coset graphs are distance transitive. In the study of the mod cohomology of the lie algebra of the unipotent radical of groups of lie type with simply laced diagram, it was found that the connected components of the hasse diagram of the koszul complex are graphs. Considering a connected graph g with diameter d, we say that it is kwalk regular, for a given integer k 0. Classification of the known distanceregular graphs.
As we shall see, the theory of distance regular graphs has connections to many parts of graph theory, design theory, coding theory, geometry both finite and euclidean, and group theory. Citeseerx document details isaac councill, lee giles, pradeep teregowda. The heawood graph has crossing number 3, and is the smallest cubic graph with that crossing number sequence a110507 in the oeis. Some introductory papers on association schemes are dfaswietf. This cited by count includes citations to the following articles in scholar. It is a strongly regular graph with 16 vertices and 48 edges, with each vertex having degree 6. Reconstruction conjecture rc asserts that all graphs on at least three vertices are reconstructible. We present an introduction to distance regular graphs for the reader who is unfamiliar with the subject, and then give an overview of some developments in the area of distance regular graphs since the monograph bcn brouwer, a. Yamada, distanceregular digraphs of girth4 over an extension ring of z4z. Tilburg university distanceregularity and the spectrum of. In this paper, it is shown that is a distance transitive graph, and, by this fact, we determine the adjacency matrix spectrum of. Euclidean representations and substructures of distance regular graphs citation for published version apa. This gives antipodal covers of some distance regular and distance transitive graphs.
A distance transitive graph is interesting partly because it has a large automorphism group. Lambeck department of malhematics and compuling science, eindhoven universiry of technology. A connected graph g is distance regular if for any nodes x,y and any integers i,j0,1. Families of nested completely regular codes and distance. This book, bcn, contained almost all information on distance regular graphs known at that moment. Some interesting finite groups are the automorphism groups of distance transitive graphs, especially of those whose diameter is 2. Users may download and print one copy of any publication from the public portal for the purpose of private study or research. A new family of distance regular graphs is constructed. There exists no distanceregular graph with intersection.
On the multiplicity of eigenvalues of distanceregular graphs. On the structure of brouwer homeomorphisms embeddable in a flow lesniak, zbigniew, abstract and applied analysis, 2012. Several other important regular combinatorial structures are then shown to be equivalent to special families of distanceregular graphs. Distanceregular graphs tilburg university research portal. This intersection array is from the table of feasible. In t roduct ion graph theory is a major area of combinatorics and during recent decades, graph theory has developed into. In this paper, we prove that intervalregular graphs and some new classes of graphs are reconstructible and show that rc is true if and only if all nongeodetic and nonintervalregular blocks g with diamg 2 or diam. In the mathematical field of graph theory, the shrikhande graph is a named graph discovered by s. For background on distance regular graphs and strongly regular graphs, we refer the reader to brouwer, cohen, and neumaier 2, brouwer and haemers 3, and cameron 4. Pdf on kwalkregular graphs cristina dalfo and ernest. Euclidean representations and substructures of distance. Zalerts allow you to be notified by email about the availability of new books according to your search query. It is proved that every nontrivial distance regular cayley graph on a dihedral.
Discrete mathematics 1 1993 2752 7ti 275 northholland note on distance regular graphs with ci b, e. Amsterdam, andries brouwer december 2010 willem haemers. A terwilliger graph is a noncomplete graph in which the intersection of the neighborhoods of any two vertices at distance 2 from each other is a. Every pair of nodes has exactly two other neighbors in common, whether the pair of nodes is connected or not. In this paper, we study the distanceregular graphs. Jan 01, 2007 the main result of this article is a classification of distance regular cayley graphs on dihedral groups. Oclcs webjunction has pulled together information and resources to assist library staff as they consider how to handle coronavirus. Brouwer, treelike spaces and related connected topological spaces, math. A family of antipodal distanceregular graphs related to. Neumaier ever since the discovery of the five platonic solids in ancient times, the study of symmetry and regularity has been one of the most fascinating aspects of mathematics.
Implementing brouwers database of strongly regular graphs nathann cohen 0 1 dmitrii v. The automorphism groups are determined, and the extended preparata codes are reconstructed using walks on these graphs. For more information on distance regular graphs, see. The ones marked may be different from the article in the profile.
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