Super connectivity of Kronecker products of graphs

Let G₁ and G₂ be two connected graphs. The Kronecker product G₁×G₂ has vertex set V(G₁×G₂)=V(G₁)×V(G₂) and the edge set . In this paper, we show that if G is a bipartite graph with κ(G)=δ(G), then G×Kn(n?3) is super-κ.     

On edge connectivity of direct products of graphs

Let λ(G) be the edge connectivity of G. The direct product of graphs G and H is the graph with vertex set V(G×H)=V(G)×V(H), where two vertices (u₁,v₁) and (u₂,v₂) are adjacent in G×H if u₁u₂∈E(G) and v₁v₂∈E(H). We prove that λ(G×Kn)=min{n(n−1)λ(G),(n−1)δ(G)} for every nontrivial graph G and n?3. We also prove that for almost every pair of graphs G and H with n vertices and edge probability p, G×H is k-connected, where k=O(²(n/logn)).     

Vertex vulnerability parameters of Kronecker products of complete graphs

Let G₁ and G₂ be two graphs. The Kronecker product G₁×G₂ of G₁ and G₂ has vertex set V(G₁×G₂)=V(G₁)×V(G₂) and edge set and v₁v₂∈E(G₂)}. In this paper, we determine some vertex vulnerability parameters of the Kronecker product of complete graphs Km×Kn for n?m?2 and n?3.     

On Kronecker products,tensor products and matrix differential calculus

The algebra of the Kronecker products of matrices is recapitulated using a notation that reveals the tensor structures of the matrices. It is claimed that many of the difficulties that are encountered in working with the algebra can be alleviated by paying close attention to the indices that are concealed beneath the conventional matrix notation. The vectorization operations and the commutation transformations that are common in multivariate statistical analysis alter the positional relationship of the matrix elements. These elements correspond to numbers that are liable to be stored in contiguous memory cells of a computer, which should remain undisturbed. It is suggested that, in the absence of an adequate index notation that enables the manipulations to be performed without disturbing the data, even the most clear-headed of computer programmers is liable to perform wholly unnecessary and time-wasting operations that shift data between memory cells.     

On maximal 3-restricted edge connectivity of regular graphs

Let G be a k-regular graph of order at least nine. It is proved in this work that graph G is maximally 3-restricted edge-connected if it is triangle-free and k>|G|/4+1, the lower bound on k is sharp to some extent. With this observation, we present an expression of the number of edge cuts in the previous graph, which may be employed to analyse the reliability of telecommunication networks.     

The super connectivity of shuffle-cubes

The shuffle-cube SQn, where , a new variation of hypercubes proposed by Li et al. [T.-K. Li, J.J.M. Tan, L.-H. Hsu, T.-Y. Sung, The shuffle-cubes and their generalization, Inform. Process. Lett. 77 (2001) 35-41], is an n-regular n-connected graph. This paper determines that the super connectivity of SQn is 2n−4 and the super edge-connectivity is 2n−2 for n?6.     

On the time required to detect cycles and connectivity in graphs

It is shown that when a graph is represented as a binary connection matrix, the problems of finding the shortest path between two nodes of a graph, of determining whether the graph has a cycle, and of determining if a graph is strongly connected each require at leastO(n ²) operations. Thus the presently known best algorithms are optimal to within a multiplicative constant.     

On m-restricted edge connectivity of undirected generalized De Bruijn graphs

An m-restricted edge cut is an edge cut of a connected graph whose removal results in components of order at least m, the minimum cardinality over all m-restricted edge cuts of a graph is its m-restricted edge connectivity. It is known that telecommunication networks with topology having larger m-restricted edge connectivity are locally more reliable for all m≤3. This work shows that if n≥7, then undirected generalized binary De Bruijn graph UB_G(2, n) is maximally m-restricted edge connected for all m≤3, where a graph G is maximally m-restricted edge connected if its m-restricted edge connectivity is equal to the minimum number of edges from any connected subgraphs S to G?S.     

On the existence and connectivity of a class of maximal planar graphs

In this work the author determines completely which Euler maximal graphs having 14 vertices of degree 5 andn−14 vertices of degree more than 5 are planar. Furthermore, a sufficient condition for such graphs to be 5-connected is also given. Partially supported by GNAFA-CNR     

The super spanning connectivity and super spanning laceability of the enhanced hypercubes

A k -container C(u,v) of a graph G is a set of k disjoint paths between u and v. A k-container C(u,v) of G is a k ^* -container if it contains all vertices of G. A graph G is k ^* -connected if there exists a k ^*-container between any two distinct vertices of G. Therefore, a graph is 1^*-connected (respectively, 2^*-connected) if and only if it is Hamiltonian connected (respectively, Hamiltonian). A graph G is super spanning connected if there exists a k ^*-container between any two distinct vertices of G for every k with 1≤k≤κ(G) where κ(G) is the connectivity of G. A bipartite graph G is k ^* -laceable if there exists a k ^*-container between any two vertices from different partite set of G. A bipartite graph G is super spanning laceable if there exists a k ^*-container between any two vertices from different partite set of G for every k with 1≤k≤κ(G). In this paper, we prove that the enhanced hypercube Q _n,m is super spanning laceable if m is an odd integer and super spanning connected if otherwise.

Super connectivity of line graphs

The super connectivity κ^′ and the super edge-connectivity λ^′ are more refined network reliability indices than connectivity κ and edge-connectivity λ. This paper shows that for a connected graph G with order at least four rather than a star and its line graph L(G), κ^′(L(G))=λ^′(G) if and only if G is not super-λ^′. As a consequence, we obtain the result of Hellwig et al. [Note on the connectivity of line graphs, Inform. Process. Lett. 91 (2004) 7] that κ(L(G))=λ^′(G). Furthermore, the authors show that the line graph of a super-λ^′ graph is super-λ if the minimum degree is at least three.     

The super connectivity of augmented cubes

The augmented cube AQn, proposed by Choudum and Sunitha [S.A. Choudum, V. Sunitha, Augmented cubes, Networks 40 (2) (2002) 71-84], is a (2n−1)-regular (2n−1)-connected graph (n≠3). This paper determines that the super connectivity of AQn is 4n−8 for n?6 and the super edge-connectivity is 4n−4 for n?5. That is, for n?6 (respectively, n?5), at least 4n−8 vertices (respectively, 4n−4 edges) of AQn are removed to get a disconnected graph that contains no isolated vertices. When the augmented cube is used to model the topological structure of a large-scale parallel processing system, these results can provide more accurate measurements for reliability and fault tolerance of the system.     

The super connectivity of exchanged hypercubes

The exchanged hypercube EH(s,t) where s?1 and t?1 are two positive integers, proposed by Loh et al. [The exchanged hypercube, IEEE Transactions on Parallel and Distributed Systems 16 (9) (2005) 866-874], is obtained by systematically removing links from a binary hypercube Qn. This paper determines that the super connectivity and the super edge-connectivity of EH(s,t) are 2s where s?t. That is, for s?t, at least 2s vertices (resp. 2s edges) of EH(s,t) are removed to get a disconnected graph that contains no isolated vertex.     

Note on the connectivity of line graphs

Let G be a connected graph with vertex set V(G), edge set E(G), vertex-connectivity κ(G) and edge-connectivity λ(G).A subset S of E(G) is called a restricted edge-cut if G−S is disconnected and each component contains at least two vertices. The restricted edge-connectivity λ₂(G) is the minimum cardinality over all restricted edge-cuts. Clearly λ₂(G)?λ(G)?κ(G).In 1969, Chartrand and Stewart have shown that

     

Component connectivity of generalized Petersen graphs

Let G be a simple non-complete graph of order n. The r-component edge connectivity of G denoted as λ_r (G) is the minimum number of edges that must be removed from G in order to obtain a graph with (at least) r connected components. The concept of r-component edge connectivity generalizes that of edge connectivity by taking into account the number of components of the resulting graph. In this paper we establish bounds of the r component edge connectivity of an important family of interconnection network models, the generalized Petersen graphs GP(n, k) in which n and k are relatively prime integers.     

相交双圈图的代数连通度的排序

简单连通图若边数等于顶点数加1,且图中所含的两个圈至少有两个公共顶点,则称该图为相交双圈图。主要给出了相交双圈图中第五到第十大代数连通度的图类。     

Super-connected but not super edge-connected graphs

A connected graph G is super-connected (resp. super edge-connected) if every minimum vertex-cut (resp. edge-cut) isolates a vertex of G. In [Super connectivity of line graphs, Inform. Process. Lett. 94 (2005) 191-195], Xu et al. shows that a super-connected graph with minimum degree at least 4 is also super edge-connected. In this paper, a characterization of all super-connected but not super edge-connected graphs is given. It follows from this result that there is a unique super-connected but not super edge-connected graph with minimum degree 3, that is, the Ladder graph L₃ of order 6, and that there are infinitely many super-connected but not super edge-connected graphs with minimum degree 1 or 2.     

Achieving good connectivity in motion graphs

Motion graphs have been widely successful in the synthesis of human motions. However, the quality of the generated motions depends heavily on the connectivity of the graphs and the quality of transitions in them. Achieving both of these criteria simultaneously though is difficult. Good connectivity requires transitions between less similar poses, while good motion quality requires transitions only between very similar poses. This paper introduces a new method for building motion graphs. The method first builds a set of interpolated motion clips, which contains many more similar poses than the original data set. The method then constructs a well-connected motion graph (wcMG), by using as little of the interpolated motion clip frames as necessary to provide good connectivity and only smooth transitions. Based on experiments, wcMGs outperform standard motion graphs across different measures, generate good quality motions, allow for high responsiveness in interactive control applications, and do not even require post-processing of the synthesized motions.