In this paper, we consider the asymptotic properties of the numbers of spanning trees and Eulerian trials in circulant digraphs and graphs. Let C(p, s1, s2, . . . , sk) be a directed or undirected circulant graph. Let T (C(p, s1, s2, . . . , sk)) and E(C(p, s1, s2, . . . , sk)) be the numbers ∗This work was partially supported by the Natural Science Foundation of China, email: fjzhang@jingxian....

A circulant C(n; S) with connection set S = {a1, a2, . . . , am} is the graph with vertex set Zn , the cyclic group of order n, and edge set E = {{i, j} : |i − j | ∈ S}. The chromatic number of connected circulants of degree at most four has been previously determined completely by Heuberger [C. Heuberger, On planarity and colorability of circulant graphs, Discrete Math. 268 (2003) 153–169]. In...

In this paper we present a distance-hereditary decomposition of optimal chordal rings of 2k nodes into a set of rings of 2k nodes, where k is the diameter. All the rings belonging to this set have the same length and their diameter corresponds to the diameter of the chordal ring in which they are embedded. The members of this embedded set of rings are non-disjoint and preserve the minimal routi...

Abstract. We show that (n, 2n) additive codes over GF(4) can be represented as directed graphs. This generalizes earlier results on self-dual additive codes over GF(4), which correspond to undirected graphs. Graph representation reduces the complexity of code classification, and enables us to classify additive (n, 2n) codes over GF(4) of length up to 7. From this we also derive classifications ...

Two spanning trees of a graph G are said to be independent if they are rooted at the same vertex r, and for each vertex v 6= r in G, the two different paths from v to r, one path in each tree, are internally disjoint. A set of spanning trees of G is independent if they are pairwise independent. A recursive circulant graph G(N, d) has N = cdm vertices labeled from 0 to N − 1, where d > 2, m > 1,...

A classical lazy random walk on cycles is known to mix to the uniform distribution. In contrast, we show that a continuous-time quantum walk on cycles exhibit strong non-uniform mixing properties. Our results include the following: • The instantaneous distribution of a quantum walk on most even-length cycles is never uniform. More specifically, we prove that a quantum walk on a cycle Cn is not ...

For n ∈ N and D ⊆ N, the distance graph P n has vertex set {0, 1, . . . , n− 1} and edge set {ij | 0 ≤ i, j ≤ n − 1, |j − i| ∈ D}. The class of distance graphs generalizes the important and very well-studied class of circulant graphs which have been proposed for numerous network applications. In view of fault tolerance and delay issues in these applications, the connectivity and diameter of cir...

The goal of this paper is two-fold. We first focus on the problem of deciding whether two monomial rotation symmetric (MRS) Boolean functions are affine equivalent via a permutation. Using a correspondence between such functions and circulant matrices, we give a simple necessary and sufficient condition. We connect this problem with the well known Ádám’s conjecture from graph theory. As applica...

The creation of a web bubble, or link farm, is a technique used by web spammers to increase the visibility of a target page T. Specialized mobile agents, called Web Marshals, are employed to detect and dismantle link farms. As the number of Web Marshals, as well as the number of hops needed to disassemble a link farm, need to be minimized, this kind of process can be seen as a graph decontamina...

In this paper we present the first polynomial time deterministic algorithm to compute the shortest path between two vertices of a circulant graph of degree four. Our spectacular algorithm only requires O(logN) bit operations, where N is the number of the vertices and it is based on shortest vector problems in a special class of lattices for L1-norm. Moreover, the technique can be extended to we...