Pancyclicity of recursive circulant graphs

In this paper, we study the existence of cycles of all lengths in the recursive circulant graphs, and we show a necessary and sufficient condition for the graph being pancyclic and bipancyclic.     

Maximum induced subgraph of a recursive circulant

The recursive circulant RC(n²,4) enjoys several attractive topological properties. Let max_?G(m) denote the maximum number of edges in a subgraph of graph G induced by m nodes. In this paper, we show that , where p₀>p₁>?>pr are nonnegative integers defined by . We then apply this formula to find the bisection width of RC(n²,4). The conclusion shows that, as n-dimensional cube, RC(n²,4) enjoys a linear bisection width.     

Reversals Cayley graphs of symmetric groups

Corresponding to genome rearrangements by reversals, we propose a new reversal Cayley graph RSn as the Cayley graph Cay(Sn,Ωn), where Sn is the symmetric group of degree n and Ωn is the set of all reversals of Sn. Graph properties including connectivity, diameter and hamiltonicity, are investigated for this kind of graphs.     

On the partition dimension of a class of circulant graphs

For a vertex v   of a connected graph G(V,E)$G(V,E)$ and a subset S of V, the distance between a vertex v and S   is defined by d(v,S)=min{d(v,x):x∈S}$d(v,S)=\min \{d(v,x):x\in S\}$. For an ordered k  -partition π={_S1,_S2…_Sk}$\pi =\{S_1,S_2\dots S_k\}$ of V, the partition representation of v with respect to π is the k  -vector r(v|π)=(d(v,_S1),d(v,_S2)…d(v,_Sk))$r(v|\pi )=(d(v,S_1),d(v,S_2)\dots d(v,S_k))$. The k-partition π is a resolving partition if the k  -vectors r(v|π)$r(v|\pi )$, v∈V(G)$v\in V(G)$ are distinct. The minimum k for which there is a resolving k-partition of V is the partition dimension of G. Salman et al. [1] in their paper which appeared in Acta Mathematica Sinica, English Series   proved that partition dimension of a class of circulant graph G(n,±{1,2})$G(n,\pm \{1,2\})$, for all even n?6$n?6$ is four. In this paper we prove that it is three.     

On the number of spanning trees of circulant graphs

In this paper, we derive a simple formula for the number of spanning trees of the circulant graphs. Some special cases of the circulant graphs are also taken into account.     

Honeycomb toroidal graphs are Cayley graphs

There is a particular family of trivalent vertex-transitive graphs that have been called generalized honeycomb tori by some and brick products by others. They have been studied as hexagonal embeddings on the torus as well. We show that all these graphs are Cayley graphs on generalized dihedral groups.     

Embedding of special classes of circulant networks,hypercubes and generalized Petersen graphs

Hypercubes are a very popular model for parallel computation because of their regularity and the relatively small number of interprocessor connections. In this paper, we present an algorithm for embedding special class of circulant networks into their optimal hypercubes with dilation 2 and prove its correctness. Also, we embed special class of circulant networks into special class of generalized Petersen graphs with dilation 2 and vice versa.     

Conditional matching preclusion for the arrangement graphs

The matching preclusion number of a graph is the minimum number of edges whose deletion results in a graph that has neither perfect matchings nor almost-perfect matchings. For many interconnection networks, the optimal sets are precisely those induced by a single vertex. Recently, the conditional matching preclusion number of a graph was introduced to look for obstruction sets beyond those induced by a single vertex. It is defined to be the minimum number of edges whose deletion results in a graph with no isolated vertices that has neither perfect matchings nor almost-perfect matchings. In this paper we find this number and classify all optimal sets for the arrangement graphs, one of the most popular interconnection networks.     

Edge-bipancyclicity and edge-fault-tolerant bipancyclicity of bubble-sort graphs

A bipartite graph G is bipancyclic if G has a cycle of length l for every even 4?l?|V(G)|. For a bipancyclic graph G and any edge e, G is edge-bipancyclic if e lies on a cycle of any even length l of G. In this paper, we show that the bubble-sort graph Bn is bipancyclic for n?4 and also show that it is edge-bipancyclic for n?5. Assume that F is a subset of E(Bn). We prove that Bn−F is bipancyclic, when n?4 and |F|?n−3. Since Bn is a (n−1)-regular graph, this result is optimal in the worst case.     

A note about some properties of BC graphs

BC graphs is an important class of hypercube-like interconnection networks. In this paper, some properties—vertex-pancyclicity, super-connectivity and the maximum number of edges joining vertices are studied.     

Linear election in pancake graphs

Pancake graphs have been proposed as an attractive alternative to hypercube networks. They have a smaller diameter and a lower degree. They also have a hierarchical structure which can be exploited in designing algorithms.In this paper, we propose a leader election algorithm for oriented pancake graphs. The algorithm has a message complexity that is linear in the order of the graph.     

The (strong) rainbow connection numbers of Cayley graphs on Abelian groups

A path in an edge-colored graph G, whose adjacent edges may have the same color, is called a rainbow path if no two edges of the path are colored the same. The rainbow connection number rc(G) of G is the minimum integer i for which there exists an i-edge-coloring of G such that every two distinct vertices of G are connected by a rainbow path. The strong rainbow connection number src(G) of G is the minimum integer i for which there exists an i-edge-coloring of G such that every two distinct vertices u and v of G are connected by a rainbow path of length d(u,v). In this paper, we give upper and lower bounds of the (strong) rainbow connection numbers of Cayley graphs on Abelian groups. Moreover, we determine the (strong) rainbow connection numbers of some special cases.     

Fault diameter of Cartesian product graphs

The (k−1)-fault diameter Dk(G) of a k-connected graph G is the maximum diameter of G−F for any F⊂V(G) with |F|<k. Krishnamoorthy and Krishnamurthy first proposed this concept and gave Dκ(G₁)+κ(G₂)(G₁×G₂)?Dκ(G₁)(G₁)+Dκ(G₂)(G₂) when κ(G₁×G₂)=κ(G₁)+κ(G₂), where κ(G) is the connectivity of G. This paper gives a counterexample to this bound and establishes Dk₁+k₂(G₁×G₂)?Dk₁(G₁)+Dk₂(G₂)+1 for any ki-connected graph Gi and ki?1 for i=1,2.     

Conditional connectivity of Cayley graphs generated by transposition trees

Given a graph G and a non-negative integer h, the Rh-(edge)connectivity of G is the minimum cardinality of a set of (edges)vertices of G, if any, whose deletion disconnects G, and every remaining component has minimum degree at least h. Similarly, given a non-negative integer g, the g-(edge)extraconnectivity of G is the minimum cardinality of a set of (edges)vertices of G, if any, whose deletion disconnects G, and every remaining component has more than g vertices. In this paper, we determine R₂-(edge)connectivity and 2-extra(edge)connectivity of Cayley graphs generated by transposition trees.     

Diagnosability of star graphs under the comparison diagnosis model

In this paper, the diagnosability of n-dimensional star graph Sn under the comparison diagnosis model has been studied. It is proved that Sn is (n−1)-diagnosable under the comparison diagnosis model when n?4.     

The pessimistic diagnosability of alternating group graphs under the PMC model

The process of identifying faulty processors is called the diagnosis of the system. Several diagnostic models have been proposed, the most popular is the PMC (Preparata, Metze and Chen) diagnostic model. The pessimistic diagnosis strategy is a classic strategy based on the PMC model in which isolates all faulty nodes within a set containing at most one fault-free node. A system is t/t$t/t$-diagnosable if, provided the number of faulty processors is bounded by t, all faulty processors can be isolated within a set of size at most t with at most one fault-free node mistaken as a faulty one. The pessimistic diagnosability of a system G  , denoted by _tp(G)$t_p(G)$, is the maximal number of faulty processors so that the system G   is t/t$t/t$-diagnosable. Jwo et al. [11] introduced the alternating group graph as an interconnection network topology for computing systems. The proposed graph has many advantages over hypercubes and star graphs. For example, for all alternating group graphs, every pair of vertices in the graph are connected by a Hamiltonian path and the graph can embed cycles with arbitrary length with dilation 1. In this article, we completely determine the pessimistic diagnosability of an n  -dimensional alternating group graph, denoted by _AGn${\mathit{AG}}_n$. Furthermore, _tp(_AGn)=4n−11$t_p({\mathit{AG}}_n)=4n-11$ for n≥4$n\ge 4$.     

Edge-bipancyclicity of star graphs with faulty elements

In this paper, we investigate the fault-tolerant edge-bipancyclicity of an n-dimensional star graph S_n. Given a set F comprised of faulty vertices and/or edges in S_n with |F|≤n−3 and any fault-free edge e in S_n−F, we show that there exist cycles of every even length from 6 to n!−2|F_v| in S_n−F containing e, where n≥3. Our result is optimal because the star graph is both bipartite and regular with the common degree n−1. The length of the longest fault-free cycle in the bipartite S_n is n!−2|F_v| in the worst case, where all faulty vertices are in the same partite set. We also provide some sufficient conditions from which longer cycles with length from n!−2|F_v|+2 to n!−2|F_v| can be embedded.     

The category-theoretic solution of recursive metric-space equations

It is well known that one can use an adaptation of the inverse-limit construction to solve recursive equations in the category of complete ultrametric spaces. We show that this construction generalizes to a large class of categories with metric-space structure on each set of morphisms: the exact nature of the objects is less important. In particular, the construction immediately applies to categories where the objects are ultrametric spaces with ‘extra structure’, and where the morphisms preserve this extra structure. The generalization is inspired by classical domain-theoretic work by Smyth and Plotkin.For many of the categories we consider, there is a natural subcategory in which each set of morphisms is required to be a compact metric space. Our setting allows for a proof that such a subcategory always inherits solutions of recursive equations from the full category.As another application, we present a construction that relates solutions of generalized domain equations in the sense of Smyth and Plotkin to solutions of equations in our class of categories.Our primary motivation for solving generalized recursive metric-space equations comes from recent and ongoing work on Kripke-style models in which the sets of worlds must be recursively defined. We show a series of examples motivated by this line of work.     

One conjecture of bubble-sort graphs

The bubble-sort graph is an important interconnection network designed from Cayley graph model. One conjecture is proposed in Shi and Lu (2008) [10] as follows: for any integer n?2, if n is odd then bubble-sort graph Bn is a union of edge-disjoint hamiltonian cycles; if n is even then bubble-sort graph Bn is a union of edge-disjoint hamiltonian cycles and its perfect matching that has no edges in common with the hamiltonian cycles. In this paper, we prove that conjecture is true for n=5,6.