Fault diagnosis of systems is an important area of study in the design and maintenance of multiprocessor systems. In 2005, Lai et al. 12 introduced conditional diagnosability under the assumption ...that all the neighbors of any processor in a multiprocessor system cannot be faulty at the same time. In this paper, we completely determine the conditional diagnosability of Cayley graphs generated by wheel graphs WGn under the PMC model.
Bakry–Émery Curvature Functions on Graphs Cushing, David; Liu, Shiping; Peyerimhoff, Norbert
Canadian journal of mathematics,
02/2020, Volume:
72, Issue:
1
Journal Article
Peer reviewed
Open access
We study local properties of the Bakry–Émery curvature function \({\mathcal{K}}_{G,x}:(0,\infty \rightarrow \mathbb{R}\) at a vertex \(x\) of a graph \(G\) systematically. Here ...\({\mathcal{K}}_{G,x}({\mathcal{N}})\) is defined as the optimal curvature lower bound \({\mathcal{K}}\) in the Bakry–Émery curvature-dimension inequality \(CD({\mathcal{K}},{\mathcal{N}})\) that \(x\) satisfies. We provide upper and lower bounds for the curvature functions, introduce fundamental concepts like curvature sharpness and \(S^{1}\)-out regularity, and relate the curvature functions of \(G\) with various spectral properties of (weighted) graphs constructed from local structures of \(G\). We prove that the curvature functions of the Cartesian product of two graphs \(G_{1},G_{2}\) are equal to an abstract product of curvature functions of \(G_{1},G_{2}\). We explore the curvature functions of Cayley graphs and many particular (families of) examples. We present various conjectures and construct an infinite increasing family of 6-regular graphs which satisfy \(CD(0,\infty )\) but are not Cayley graphs.
The family of finite 2-arc-transitive graphs of a given valency is closed under forming non-trivial normal quotients, and graphs in this family having no non-trivial normal quotient are called ...‘basic’. To date, the vast majority of work in the literature has focused on classifying these ‘basic’ graphs. By contrast we give here a characterisation of the normal covers of the ‘basic’ 2-arc-transitive graphs K2n,2n for n≥2. The characterisation identified the special role of graphs associated with a subgroup of automorphisms called an n-dimensional mixed dihedral group. This is a group H with two subgroups X and Y, each elementary abelian of order 2n, such that X∩Y=1, H is generated by X∪Y, and H/H′≅X×Y.
Our characterisation shows that each 2-arc-transitive normal cover of K2n,2n is either itself a Cayley graph, or is the line graph of a Cayley graph of an n-dimensional mixed dihedral group. In the latter case, we show that the 2-arc-transitive group acting on the normal cover of K2n,2n induces an edge-affine action on K2n,2n (and we show that such actions are one of just four possible types of 2-arc-transitive actions on K2n,2n). As a partial converse, we provide a graph theoretic characterisation of n-dimensional mixed dihedral groups, and finally, for each n≥2, we give an explicit construction of an n-dimensional mixed dihedral group which is a 2-group of order 2n2+2n, and a corresponding 2-arc-transitive normal cover of 2-power order of K2n,2n. Note that these results partially address a problem proposed by Caiheng Li concerning normal covers of prime power order of the ‘basic’ 2-arc-transitive graphs.
A subset F⊂V(G) is called an Rk-vertex-cut if G-F is disconnected and each vertex u∈V(G)-F has at least k neighbors in G-F. The cardinality of a minimum Rk-vertex-cut is the Rk-vertex-connectivity of ...G and is denoted by κk(G). The conditional connectivity is a measure to study the structure of networks beyond connectivity. Hypercubes form the basic classes of interconnection networks. Complete transposition graphs were introduced to be competitive models of hypercubes. In this paper, we determine the numbers κ1 and κ2 for complete-transposition graphs, κ1(CTn)=n(n-1)-2,κ2(CT4)=16 and κ2(CTn)=2n(n-1)-10 for n⩾5.
Abstract
Let
$\Gamma $
be a graph of valency at least four whose automorphism group contains a minimally vertex-transitive subgroup
G
. It is proved that
$\Gamma $
admits a nowhere-zero
$3$
-flow if ...one of the following two conditions holds: (i)
$\Gamma $
is of order twice an odd number and
G
contains a central involution; (ii)
G
is a direct product of a
$2$
-subgroup and a subgroup of odd order.
In 2011, Fang et al. in 9 posed the following problem: Classify non-normal locally primitive Cayley graphs of finite simple groups of valency d, where eitherd≤20or d is a prime number. The only case ...for which the complete solution of this problem is known is of d=3. Except this, a lot of efforts have been made to attack this problem by considering the following problem: Characterize finite nonabelian simple groups which admit non-normal locally primitive Cayley graphs of certain valencyd≥4. Even for this problem, it was only solved for the cases when either d≤5 or d=7 and the vertex stabilizer is solvable. In this paper, we make crucial progress towards the above problems by completely solving the second problem for the case when d≥11 is a prime and the vertex stabilizer is solvable.
The isomorphism problem is a fundamental problem for algebraic and combinatorial structures, particularly in relation to Cayley graphs. Let Xi=GC(G,Si,αi),(i=1,2) be generalized Cayley graphs. If ...whenever X1≅X2, it implies that α2=α1γ and S2=g−1S1γgα2 for some g∈G and γ∈Aut(G), then G is a strongly generalized Cayley isomorphism (GCI)-group. In this study, we defined (strongly, restricted) m-GCI-groups. These definitions are similar to those of m-CI-groups for Cayley graphs. Our main results demonstrate that a finite non-abelian simple group G is a restricted 2-GCI-group if and only if G is one of A5, L2(8), M11, Sz(8), or M23, and G is a 2-GCI-group if and only if G is A5 or L2(8).
Tutte's 3-flow conjecture asserts that every 4-edge-connected graph admits a nowhere-zero 3-flow. We prove that this conjecture is true for every Cayley graph of valency at least four on any ...supersolvable group with a noncyclic Sylow 2-subgroup and every Cayley graph of valency at least four on any group whose derived subgroup is of square-free order.
•Neighbor connectivity κNB measures the fault-tolerance of interconnection networks.•We completely determine κNB for the class of Cayley graphs formed by k-ary n-cubes.•These graphs reach the maximum ...value of κNB conjectured by Doty in 2006.
The neighbor connectivity of a graph G is the least number of vertices such that removing their closed neighborhoods from G results in a graph that is disconnected, complete or empty. If a graph is used to model the topology of an interconnection network, this means that the failure of a network node causes failures of all its neighbors. We completely determine the neighbor connectivity of k-ary n-cubes for all n ≥ 1 and k ≥ 2.
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