Euclidean representations and substructures of distance. 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. 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. Improving diameter bounds for distanceregular graphs. Reliable information about the coronavirus covid19 is available from the world health organization current situation, international travel. Distanceregular graphs the bannaiito conjecture geometric drg open problems and conjectures brouwer, cohen and neumaier asked whether every drg with valency at least three and diameter at least three has an integral eigenvalue besides its valency. Some algebraic and spectral properties of distance meanregular graphs are also investigated. Two distanceregular graphs two distanceregular graphs brouwer, andries. A connected graph g is distanceregular if for any nodes x,y and any integers i,j0,1. The rst is the connection between bipartite distanceregular graphs of diameter four and strongly regular graphs. Returns true if the graph is distance regular, false otherwise. 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. Conways 99graph problem asks for the construction of an srg99, 14, 1, 2. Oclcs webjunction has pulled together information and resources to assist library staff as they consider how to handle coronavirus.
Finally, we refer the reader who has become interested in the topic to recent surveys by fiol 11, 12. Cohen and neumaier 4, and cvetkovic, doob and sachs 7, as basic references. This intersection array is from the table of feasible. A strongly regular graph is called primitive if both the graph and its complement are connected. Pdf parameter restrictions for distanceregular graphs. This intersection array is from the table of feasible parameters for distanceregular graphs in distanceregular graphs\ by a. Amsterdam, andries brouwer december 2010 willem haemers. The average distance in a random graph with given expected degrees chung. 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.
A reduction of the graph reconstruction conjecture in. Brouwer 93 constructed related distanceregular graphs with. A table of parameters of directed strongly regular graphs. Merging the first and third classes in bipartite distance. For several topics on algebraic combinatorics, such as distanceregular graphs and orthogonal polynomials, the book by godsil 15 will be useful. Distanceregular cayley graphs on dihedral groups, journal. On walkregular graphs and graphs with symmetric hitting times. On vertex decomposable and cohen macaulay regular graphs luviano, j. By looking at the eigenvalues of distanceregular graphs it is. Several other important regular combinatorial structures are then shown to be equivalent to special families of distanceregular graphs. Families of distanceregular graphs with large girth. In pyber showed that the diameter of distanceregular graphs is at most 5 times the 2 logarithm of the number of vertices.
Neumaier, introduction to numerical analysis, cambridge univ. If this is the first time you use this feature, you will be asked to authorise cambridge core to connect with your account. On the structure of brouwer homeomorphisms embeddable in a flow lesniak, zbigniew, abstract and applied analysis, 2012. Neumaier, a unified view of inequalities for distanceregular graphs, part i, manuscript 2018. Including the heawood graph, there are 8 distinct graphs of order 14 with crossing number 3.
Neumaier, efficient global unconstrained black box optimization, manuscript 2019. Smiths theorem and a characterization of the 6cube as. The source is some troff dialect, with most formatting commands removed. Thus, i first learned the construction for the gewirtz graph from this book. Anyone working on the subject probably knows the books of biggs 3, brouwer. The main result of this paper is that the distanceregular graphs described in the.
This book, bcn, contained almost all information on distance regular graphs known at that moment. In 8, we proved a generalized version of above result. There exist four obvious families of such graphs, which are called trivial. The distanceregular graphs with valencies 3 and 4 are completely classi.
Frederic vanhove pointed out a mistake in theorem 12. Distancetransitive graphs were first defined in 1971 by norman l. Although we shall develop large parts of the theory of distanceregular graphs independently of chapter 2, we shall use concepts and results about association schemes for more specialized topics such as qpolynomial orderings chapter 8 and codes in graphs chapter 11. It is shown that a distanceregular graph with classical parameters has the qpolynomial property 2, theorem 8. New distance regular graphs arising from dimensional dual. Neumaier, a unified view of inequalities for distanceregular graphs, part ii, manuscript 2018. Brouwer, cohen, neumaier proved that a qpolynomial regular near polygon with diameter d. Distanceregular graph doesnt have graph embeddings, but it has the constructions for hundreds and hundreds of famous graphs.
The heawood graph has crossing number 3, and is the smallest cubic graph with that crossing number sequence a110507 in the oeis. 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. The first section may be viewed as a short introduction to the subject. Using this we show that for distanceregular graphs with certain intersection arrays, the first subconstituent graphs are strongly regular. Contributions to the theory of distance regular graphs. Nonexistence of some antipodal distanceregular graphs of. The reader is referred to brouwercohenneumaier 5 and van damkoolen tanaka 7 for more background information on distance regular. There exists no distanceregular graph with intersection. The central problem in the theory of distance regular graphs is their classification, which seems to be very hard. 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.
New distance regular graphs arising from dimensional dual hyperovals antonio pasini and satoshi yoshiara in 4. In the mathematical field of graph theory, a distancetransitive graph is a graph such that, given any two vertices v and w at any distance i, and any other two vertices x and y at the same distance, there is an automorphism of the graph that carries v to x and w to y. It is known that the classical distanceregular graphs contain many nice substructures, like designs in the johnson schemes and the binary golay code in the 23cuhe. Distanceregular graphs brouwer cohen neumaier 1988. Distanceregular graphs are graphs with a lot of combinatorial symmetry, in the sense that given an arbitrary ordered pair of vertices at distance h, the number of vertices that are at distance ifrom the rst vertex and distance jfrom the second is a constant i. Starting from very elementary regularity properties, the concept of a distanceregular graph arises naturally as a common setting for regular graphs which are extremal in one sense or another. Distanceregular graphs the electronic journal of combinatorics. 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.
Some instances of distance mean regular graphs are the. For distanceregular graphs, see the book by brouwer, cohen, and neumaier 1. Algebraic graph theory is the branch of mathematics that studies graphs by using. In this paper, we consider a qpolynomial distanceregular graph with a 1 0,a 2. Some algebraic and spectral properties of distance mean regular graphs are also investigated. Arnold 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. Also, we will classify shilla distanceregular graphs with b. I learned about the construction of the perkel graph, and realized it was the same as the 57cell skeleton. Furthermore, we will give a new existence condition for distanceregular graphs, in general. The spectral excess theorem for distanceregular graphs. This cited by count includes citations to the following articles in scholar.
We prove that a distanceregular graph with intersection array 56,36,9. The primitive distancetransitive representations of the fischer groups linton, stephen a. Distanceregular cayley graphs on dihedral groups distanceregular cayley graphs on dihedral groups miklavic stefko. Numerous and frequentlyupdated resource results are available from this search. The reader is referred to brouwer cohen neumaier 5 and van damkoolentanaka 7 for more background information on distance regular graphs. The ones marked may be different from the article in the profile. We present an introduction to distanceregular graphs for the reader who is. Are there known families of distanceregular graphs with girth larger than 4 where for given vertexedge count there are more than one nonisomorphic instances. On walkregular graphs and graphs with symmetric hitting times agelos georgakopoulos. Mathematics institute university of warwick cv4 7al, uk abstract aldous 1 asked whether every graph in which the distribution of the return time of random is independent of the starting vertex must be transitive.
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