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