12,702 research outputs found
Simple random walk on distance-regular graphs
A survey is presented of known results concerning simple random walk on the
class of distance-regular graphs. One of the highlights is that electric
resistance and hitting times between points can be explicitly calculated and
given strong bounds for, which leads in turn to bounds on cover times, mixing
times, etc. Also discussed are harmonic functions, moments of hitting and cover
times, the Green's function, and the cutoff phenomenon. The main goal of the
paper is to present these graphs as a natural setting in which to study simple
random walk, and to stimulate further research in the field
A Sharp upper bound for the spectral radius of a nonnegative matrix and applications
In this paper, we obtain a sharp upper bound for the spectral radius of a
nonnegative matrix. This result is used to present upper bounds for the
adjacency spectral radius, the Laplacian spectral radius, the signless
Laplacian spectral radius, the distance spectral radius, the distance Laplacian
spectral radius, the distance signless Laplacian spectral radius of a graph or
a digraph. These results are new or generalize some known results.Comment: 16 pages in Czechoslovak Math. J., 2016. arXiv admin note: text
overlap with arXiv:1507.0705
A characterization of testable hypergraph properties
We provide a combinatorial characterization of all testable properties of
-graphs (i.e. -uniform hypergraphs). Here, a -graph property
is testable if there is a randomized algorithm which makes a
bounded number of edge queries and distinguishes with probability between
-graphs that satisfy and those that are far from satisfying
. For the -graph case, such a combinatorial characterization was
obtained by Alon, Fischer, Newman and Shapira. Our results for the -graph
setting are in contrast to those of Austin and Tao, who showed that for the
somewhat stronger concept of local repairability, the testability results for
graphs do not extend to the -graph setting.Comment: 82 pages; extended abstract of this paper appears in FOCS 201
The Resolving Graph of Amalgamation of Cycles
For an ordered set W = {w_1,w_2,...,w_k} of vertices and a vertex v in a connected graph G, the representation of v with respect to W is the ordered k-tuple r(v|W) = (d(v,w_1),d(v,w_2),...,d(v,w_k)) where d(x,y) represents the distance between the vertices x and y. The set W is called a resolving set for G if every vertex of G has
a distinct representation. A resolving set containing a minimum number of vertices is called a basis for G. The dimension of G, denoted by dim(G), is the number of vertices in a basis of G. A resolving set W of G is connected if the subgraph induced by W is a nontrivial connected subgraph of G. The connected resolving number is the minimum cardinality of a connected resolving set in a
graph G, denoted by cr(G). A cr-set of G is a connected resolving set with cardinality cr(G). A connected graph H is a resolving graph if there is a graph G with a cr-set W such that = H. Let {G_i} be a finite collection of graphs and each G_i has a fixed vertex v_{oi} called a terminal. The amalgamation Amal{Gi,v_{oi}} is formed by
taking of all the G_i's and identifying their terminals. In this paper, we determine the connected resolving number and characterize the resolving graphs of amalgamation of cycles
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