Parallelogramfree distanceregular graphs yuhjeng liang and chihwen weng department of applied mathematics, national chiao tung university, taiwan, republic of china received. Interpretation of motion graphs page 1 of 2 fdhs physics distancetime graph a distancetime graph tells us how far an object has moved with time, using the fundamental motion equation. Bcn tables in bcn tables for feasible parameter sets for distanceregular graphs on at most 4096 vertices were given. Distanceregular subgraphsin a distanceregular graph, i. If this is the first time you use this feature, you will be asked to authorise cambridge core to connect with your account. Let g x and, for x and y in x, let be adjacent to y g if only is of. An overview medhaitagihuilgol department of mathematics, bangalore university, central college. 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. We classify triangle and pentagonfree distanceregular graphs with diameter d. 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 open problems characterizations about our proof distancetransitive graphs distanceregular graphs were introduced by biggs in the late 1960s as a combinatorial. Simply explained alexander coulter paauwe april 20, 2007. There is no closed formula that anyone knows of, but there are asymptotic results, due to bollobas, see a probabilistic proof of an asymptotic formula for the number of labelled regular. The corresponding tables for the case of bipartite graphs of diameter 4 were given in the pdf. We give necessary and sufficient conditions for the neighbourhood. Department of applied mathematics national chiao tung university taiwan. The distance degree sequence dds of a vertex v in a graph g v, e is a list of the number of vertices at distance 1, 2. Pdf distance degree regular graphs and distance degree. Read parallelogramfree distanceregular graphs, journal of combinatorial theory, series b on deepdyve, the largest online rental service for scholarly research with thousands of academic. Special regular graphs association schemes representation theory theory of distanceregular graphs parameter restrictions for distanceregular graphs classification of the known. The concept of distance pervades all of science and mathematics, and even our daily lives.
Distance time graphs step by step worksheet differentiated. Sometimes graphs help make motion easier to picture, and. A new family of distanceregular graphs with unbounded. Part ii create a distance vs time graph 1 a helicopter left the landing pad at the top of a skyscraper and then quickly flew downwards towards the ground and maintained a 5 foot. In particular, we prove that such a graph is isomorphic to a. A survey is presented of known results concerning simple random walk on the class of distanceregular graphs. Research article distance in graph theory and its application mahesh c. Then we will look at distanceregular graphs, determine for all known ones whether they are 3chromatic, and make a start with the classification of all 3chromatic distanceregular graphs. On vertex decomposable and cohenmacaulay regular graphs luviano, j.
The distanceregular graphs with diameter 2 are very special, and form a subject of their own. This paper gives an introduction to the area of graph theory dealing with properties of regular graphs of given girth. 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. One of the highlights is that electric resistance and hitting times between. Parallelogramfree distanceregular graphs sciencedirect. Review article distance degree regular graphs and distance.
Starting from very elementary regularity properties, the concept of a distanceregular graph. Pdf triangle and pentagonfree distanceregular graphs. Name date period distancetime graphs describing the motion of an object is occasionally hard to do with words. Nonetheless, there are many constructions of such graphs in the literature 26, 41, 42, most notably those derived from distanceregular graphs 9 and from association schemes. The wellknown petersen graph is a distanceregular graph with diameter 2, and has in.
Weaker students do not need to complete each step before moving on to the next question. In this paper we prove that there are finitely many trianglefree distanceregular graphs with degree 8, 9 or 10. Since all distance regular graphs are transmission regular graphs, by lemma 9, equality occurs in theorems 5 and 6 for each of the graphs. Arnold neumaier ever since the discovery of the five platonic solids in ancient times, the study of. Distanceregular graphs and halved joe hemmeter let g be a bipartite distanceregular graph with bipartition v g xu y. Mathematics free fulltext on the generalized distance. Line graphs are used to track changes over short and long periods of time.
A separate survey of strongly regular graphs would therefore be warmly welcomed. This interplay between regularity and symmetry properties of graphs is the theme of this book. In particular, we prove that such a graph is isomorphic to a cycle, a kcube, a complete bipartite graph minus a. Bipartite distanceregular graphs with an eigenvalue of multiplicity equal to the valency have alreadybeenclassi.
A large portion of the paper is based on exercises and questions. We believe that to find nice distanceregular subgraphs in a distanceregular graph will be a key for the classification of distanceregular graphs. In mathematics, a distanceregular graph is a regular graph such that for any two vertices v and w, the number of vertices at distance j from v and at distance k from w depends only upon j, k, and. The big project of classifying distanceregular graphs. Several other important regular combinatorial structures are then shown to be equivalent to special families of distanceregular graphs. Distanceregular graphs 5 two case, but we are of the opinion that they form a subject of their own. Folge a series of modern surveys in mathematics on free shipping on qualified orders. A questionable distanceregular graph rebecca ross abstract in this paper, we introduce distanceregular graphs and develop the intersection algebra for these graphs which is based. We study the relation between distanceregular graphs and. When smaller changes exist, line graphs are better to use than bar graphs. Permission is granted to copy, distribute andor modify this document under the terms of the gnu free.
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