3-Restricted Edge Connectivity of Vertex Transitive Graphs of Girth Three

Let G be a k-regular connected graph of order at least six. If G has girth three, its 3-restricted edge connectivity λ3(G) ≤3k-6. The equality holds when G is a cubic or 4-regular connected vertex-transitive graph with the only exception that G is a 4-regular graph with λ3(G) = 4. Furthermore, λ3(G) = 4 if and only if G contains K4 as its subgraph.

Bound on m-restricted Edge Connectivity

An m-restricted edge cut is an edge cut that separates a connected graph into a disconnected one with no components having order less than m. m-restricted edge connectivity λ m is the cardinality of a minimum m-restricted edge cut. Let G be a connected k-regular graph of order at least 2m that contains m-restricted edge cuts and X be a subgraph of G. Let ∂(X) denote the number of edges with one end in X and the other not in X and ξ m = min{∂(X) : X is a connected vertex-induced subgraph of order m}. It is proved in this paper that if G has girth at least m/2+ 2, then λ m ≤ ξ m . The upper bound of λ m is sharp.

Conditional Edge Connectivity of the Locally Twisted Cubes

The k-component edgeconnectivity cλk(G)of anon-completegraph G is themini mum number of edges whose deletion results in a graph with at least k components In this paper,we extend some results by Guo et al.(Appl Math Comput 334:401-406,2018)by determining the component edge connectivity of the locally twisted cubes LTQn,i.e.,cλk+1(LTQn)=kn-exk/2for 1≤2[n/2],n≥7,where exk=∑si=0ti2ti+∑si=02·i·2ti,and k is a positive integer with decomposition k=∑si=02ti such that to="log2k"and ti="log2(k-∑i-1 r=02tr)"for i≥1.As a by-product,we characterize the corresponding optimal solutions.

On Edge Connectivity and Parity Factor

By Petersen＇s Theorem, a bridgeless cubic graph has a 2-factor. Fleischner （Discrete Math., 101, 33-37 （1992）） has extended this result to bridgeless graphs of minimum degree at least three by showing that every such graph has an even factor without isolated vertices. Let me〉 0 be even and mo〉 0 be odd. In this paper, we prove that every me-edge-connected graph with minimum degree at least me ＋ 1 contains an even factor with minimum degree at least me and every （mo ＋ 1）- edge-connected graph contains an odd factor with minimum degree at least too, which further extends Fleischner＇s result. Moreover, we show that our results are best possible.

ON THE CONDITIONAL EDGE CONNECTIVITY OF ENHANCED HYPERCUBE NETWORKS

Let G =（V, E） be a connected graph and m be a positive integer, the conditional edge connectivity λ;is the minimum cardinality of a set of edges,if it exists, whose deletion disconnects G and leaves each remaining component with minimum degree δ no less than m. This study shows that λ;(Q;) = 2 n,λ;(Q;) = 4 n-4（2 ≤ k ≤ n-1, n ≥ 3） for n-dimensional enhanced hypercube Q;. Meanwhile, another easy proof about λ;（Q;） = 4 n-8, for n ≥ 3 is proposed. The results of enhanced hypercube include the cases of folded hypercube.

The Edge Connectivities of Total P_(k)-graphs

The P_(k)-path graph P_(k)(G)corresponding to a graph G has for vertices the set of all paths of length k in G.Two vertices are joined by an edge if and only if the intersection of the corresponding paths forms a path of length k-1 in G,and their union forms either a cycle or a path of length k+1.Let Ek={(v,p),p E V(P_(k)(G)),v is an end vertex of p in G},we define total P_(k)-graphs T_(k)(G)as Yk(G)=(V(G)UV(P_(k)(G)),E(G)U E(PI(G))U Ek).In this note,we introduce total P,-graphs Th(G)and study their edge connectivity,as the generaliza-tion of total graphs.

On the Decay Number of a Graph

The decay number of a connected graph is defined to be the minimum number of the components of the cotree of the graph. Upper bounds of the decay numbers of graphs are obtained according to their edge connectivities. All the bounds in this paper are tight.Moreover, for each integer k between one and the upper bound, there are infinitely many graphs with the decay number k.

Super-Edge-Connectivity of G(k,d,s)(s≥k/2)

A graph G is super-edge-connected,for short super-λ,if every minimum edge-cut consists of edges adjacent to a vertex of minimum degree.Alphabet overlap graph G（k,d,s）is undirected,simple graph with vertex set V={v｜v=1（）kv…v;vi∈{1,2,…,d},i=1,…,k}.Two vertices u=（u1…uk）and v=（v1…vk）are adjacent if and only if us＋i=vi or vs＋i=ui（i=1,…,k-s）.In particular G（k,d,1）is just an undirected de Bruijn graph.In this paper,we show that the diameter of G（k,d,s）is k s,the girth is 3.Finally,we prove that G（k,d,s）（s≥k/2）is super-λ.

REFINED CONNECTIVITY PROPERTIES OF ABELIAN CAYLEY GRAPHS

Restricted edge connectivity of a graph G is defined to be the minimum size |U| of a set U of edges such that G-U is disconnected and G-U contains no trivial component K1. The high order edge connectivity Ni, i1, is the number of edge outsets of size i. TO determine all Ni, i 1, for a general graph is NP-hard. In this paper, the authors evaluated the restricted edge connectivity and the high order edge connectivity Ni, 1 i -1, for any connected Abelian Cayley graphs explicitly.

Neighbor Connectivity of Two Kinds of Cayley Graphs

In this paper, we determine the neighbor connectivity κNB of two kinds of Cayley graphs： alter- nating group networks ANn and star graphs Sn; and give the exact values of edge neighbor connectivity λNB of ANn and Cayley graphs generated by transposition trees Fn. Those are κNB（ANn） = n-1, λNB（ANn） = n-2 and κNB（Sn） = λNB（Гn） = n - 1.

Residual Closeness of Graphs with Given Parameters

Robustness of the network topology is a key aspect in the design of computer networks. Residual closeness is a new graph-theoretic concept defined as a measure of network robustness due to the failure of individual vertices. We identify those graphs with maximum residual closeness among connected graphs with fixed connectivity, edge connectivity and bipartiteness, respectively.