If g is regular, then this return time is just n, the number of nodes. We use this name informally for graphs that share some regularity properties that are related to distance in the graph. Having established this connection, we construct new examples of regular 3 graphs via antipodal distance regular covers of complete graphs. The rst is the connection between bipartite distanceregular graphs of diameter four and strongly regular graphs. This interplay between regularity and symmetry properties of graphs is the theme of this book. Ddi graphs are highly irregular, in comparison with the ddr graphs. Pdf this report considers the resistance distance as a recently proposed new intrinsic metric on molecular graphs, and in particular, the sum r over. Resistance distance in regular graphs lukovits 1999.
However, some authors use oriented graph to mean the same as directed graph. Pdf algebraic characterizations of distanceregular graphs. The latter is the extended bipartite double of the former. We construct two families of distanceregular 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. A separate survey of strongly regular graphs would therefore be warmly welcomed. A 0 regular graph consists of disconnected vertices, a 1 regular graph consists of disconnected edges, and a 2 regular graph consists of a disjoint union of cycles and infinite chains. Starting from very elementary regularity properties, the concept of a distance regular graph arises naturally as a common setting for regular graphs. Consider a connected simple graph with vertex set x of diameter d.
Displaying distance vs time graphs worksheet with answers. In chapter 2, we study the theory of representations of distance regular graphs, which is the main machinery throughout this thesis. The classification of distanceregular cayley graphs is an open problem in the area of algebraic graph theory 28. We therefore study the problem of which of these distance regular graphs with small valency are cayley graphs. Regular graphs of degree at most 2 are easy to classify. Weakgeodetically closed subgraphs in distanceregular graphs.
A graph is distance degree injective ddi graph if no two vertices have the same distance degree sequence. Department of mathematics, royal holloway and bedford new college, egham, surrey tw20 0ex. In pyber showed that the diameter of distance regular graphs. The concept of distance degree regular ddr graphs was introduced by bloom et al. Regular tgraphs, antipodal distance regular graphs of. Thus, if a is a distance regular graph with adjacency matrix a, then each distance matrix ai u,a where uix. On the moorepenrose inverse of distanceregular graphs. A characterization of qpolynomial distanceregular graphs. Dbounded distance regular graphs 2 by abuse of notation, we refer to this subgraph as d. Although we shall develop large parts of the theory of distanceregular graphs. Available formats pdf please select a format to send. Pdf resistance distance in regular graphs researchgate. Distanceregular graphs 5 two case, but we are of the opinion that they form a subject of their own.
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. After some preliminaries in section 2, we study several families of distance. These graphs are generalizations of the strongzy regular graphs, which are just distance regular graphs. Pdf on almost distanceregular graphs cristina dalfo. On the distance spectra of graphs ghodratollah aalipour. Distance regular graphs of qracah type and the qtetrahedron algebra tatsuro ito. That is, it is a directed graph that can be formed as an orientation of an undirected graph. Distance degree regular graphs and distance degree injective graphs. 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. Distance degree regular graphs and distance degree. A survey 5 visited this node, then the expected number of steps before it returns is 1i 2mdi.
An oriented graph is a directed graph in which at most one of x, y and y, x may be edges of the graph. Shortest paths in distanceregular graphs semantic scholar. Strongly regular graphs are elusive and somewhat mysterious objects that have connections to various combinatorial constructions and to algebra over. A new family of distanceregular graphs with unbounded. Pencil, pen, ruler, protractor, pair of compasses and eraser you may use tracing paper if needed guidance 1. A couple of slightly more complicated general families are described below. Several other important regular combinatorial structures are then shown to be equivalent to special families of distanceregular graphs. This provides the first known family of nonvertextransitive distanceregular graphs. By definition, a ddr graph must be a regular graph, but a regular graph. For example, a known characterization by rowlinson 24 of a distance regular graph. The y alency k x of a vertex x p x is the cardinality of g 1 x. The first section may be viewed as a short introduction to the subject.
Read each question carefully before you begin answering it. On the moorepenrose inverse of distance regular graphs e. An important property of distance regular graphs involves the distance matrices defined earlier. Bipartite graphs and problem solving jimmy salvatore university of chicago august 8, 2007. It is shown that there are just thirteen finite graphs which are cubic regular with valency three and distance. Bcn, contained almost all information on distance regular graphs known at that moment. Contributions to the theory of distance regular graphs. The distinct cubic distanceregular graphs are k 4 or tetrahedron, k 3,3, the petersen graph, the cube, the heawood graph, the pappus graph, the coxeter graph, the tuttecoxeter graph, the dodecahedron. X,r denote a distance regular graph with diameter d. 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. Improving diameter bounds for distanceregular graphs. For the basic theory of these graphs see biggs 3,4. By using this service, you agree that you will only keep articles for personal use, and will not openly.
Distanceregular graph, shortest path, equilibrium potential, capacity. One of the main questions of the theory of distanceregular graphs is for a given intersection array to construct a distance regular graph. Many distance regular graphs are known to be determined by their parameters, and some of these are also determined by their spectrum see section 6. Abstract through this thesis we introduce distanceregular graphs, and present some of their characterizations which depend on information retrieved from their.
Pdf distance degree regular graphs and distance degree. Some tools we assume familiarity with basic results from linear algebra, graph. As a natural generalization of distanceregular graphs see 3, 4 for the theory of distanceregular graphs, wang and suzuki 6 introduced the concept of weakly distance regular digraphs. Taut distanceregular graphs and the subconstituent algebra. Define ri x2 by x, y ri whenever x and y have graph distance. It has been proved that rg nrk n, where g n denotes a connected graph containing n vertices and k n denotes a complete graph. It has to satisfy numerous feasiblity conditions e. For more background information we refer the reader. Review article distance degree regular graphs and distance.
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