Cartesian Product Of Complete Graphs Research Articles

完全图的笛卡尔积的广义3-连通度

完全图的笛卡尔积的广义3-连通度

Distinguishing chromatic numbers of complements of Cartesian products of complete graphs

The distinguishing chromatic number of a graph, G, is the minimum number of colours required to properly colour the vertices of G so that the only automorphism of G that preserves colours is the identity. There are many classes of graphs for which the distinguishing chromatic number has been studied, including Cartesian products of complete graphs (Jerebic and Klavžar, 2010). In this paper we determine the distinguishing chromatic number of the complement of the Cartesian product of complete graphs, providing an interesting class of graphs, some of which have distinguishing chromatic number equal to the chromatic number, and others for which the difference between the distinguishing chromatic number and chromatic number can be arbitrarily large.

Decomposition of Product Graphs into Paths and Cycles of Length Four

Let $$P_{k}$$Pk and $$C_k$$Ck respectively denote a path and a cycle on $$k$$k vertices. In this paper we give necessary and sufficient conditions for the existence of $$\{P_{5}, C_4\}_{\{p, q\}}$${P5,C4}{p,q}-decomposition of tensor product and cartesian product of complete graphs. Further we extent such decomposition to complete multipartite graphs.

P3-factorization of triangulated Cartesian product of complete graphs

Partition of G into edge-disjoint H-factors is called an H-factorization of G. Shehzad Afzal and Clemens Brand have factorized graphs into factors which are isomorphic to triangulated Cartesian product of two subgraphs. Also they have discussed properties of triangulated Cartesian product of more than two graphs. In this paper, we show that the necessary conditions mn ≡ 0 (mod 3), m, n are odd and 3(mn + m + n - 3) ≡ 0 (mod 8) are sufficient for the existence of a P3-factorization of Km ⧅ Kn, where ⧅ denotes triangulated Cartesian product of graphs, if one of the following holds: (i) m ≡ 9 (mod 12), n = 5, 13 and 17, (ii) m ≡ 9 (mod 12), ns, s > 1, n = 5, 13 and 17, (iii) m ≡ 9 (mod 12), n = psqt for all s, t ≥ 1, where p = 5, 13 and q = 13, 17, p ≠ q, (iv) m ≡ 9 (mod 12), n ≡ 9 (mod 12).

Topological Characterization of Hamming and Dragonfly Networks and Its Implications on Routing

Current High-Performance Computing (HPC) and data center networks rely on large-radix routers. Hamming graphs (Cartesian products of complete graphs) and dragonflies (two-level direct networks with nodes organized in groups) are some direct topologies proposed for such networks. The original definition of the dragonfly topology is very loose, with several degrees of freedom, such as the inter- and intragroup topology, the specific global connectivity, and the number of parallel links between groups (or trunking level). This work provides a comprehensive analysis of the topological properties of the dragonfly network, providing balancing conditions for network dimensioning, as well as introducing and classifying several alternatives for the global connectivity and trunking level. From a topological study of the network, it is noted that a Hamming graph can be seen as a canonical dragonfly topology with a high level of trunking. Based on this observation and by carefully selecting the global connectivity, the Dimension Order Routing (DOR) mechanism safely used in Hamming graphs is adapted to dragonfly networks with trunking. The resulting routing algorithms approximate the performance of minimal, nonminimal, and adaptive routings typically used in dragonflies but without requiring virtual channels to avoid packet deadlock, thus allowing for lower cost router implementations. This is obtained by properly selecting the link to route between groups based on a graph coloring of network routers. Evaluations show that the proposed mechanisms are competitive with traditional solutions when using the same number of virtual channels and enable for simpler implementations with lower cost. Finally, multilevel dragonflies are discussed, considering how the proposed mechanisms could be adapted to them.

About some robustness and complexity properties of [formula omitted]-graphs networks

Given a finite group G and a set S⊂G, we consider the different cosets of each cyclic group 〈s〉 with s∈S. Then the G-graphΦ(G,S) associated with G and S can be defined as the intersection graph of all these cosets. These graphs were introduced in Bretto and Faisant (2005) as an alternative to Cayley graphs: they still have strong regular properties but a more flexible structure. We investigate here some of their robustness properties (connectivity and vertex/edge-transitivity) recognized as important issues in the domain of network design. In particular, we exhibit some cases where G-graphs are optimally connected, i.e. their edge and vertex-connectivity are both equal to the minimum degree. Our main result concerns the case of a G-graph associated with an abelian group and its canonical base S˜, which is shown to be optimally connected. We also provide a combinatorial characterization for this class as clique graphs of Cartesian products of complete graphs and we show that it can be recognized in polynomial time. These results motivate future researches in two main directions: revealing new classes of optimally connected G-graphs and investigating the complexity of their recognition.

Labeling amalgamations of Cartesian products of complete graphs with a condition at distance two

The spectrum allocation problem in wireless communications can be modeled through vertex labelings of a graph, wherein each vertex represents a transmitter and edges connect vertices whose corresponding transmitters are operating in close proximity. One well-known model is the L(2,1)-labeling of a graph G in which a function f maps the vertices of G to the nonnegative integers such that if vertices x and y are adjacent, then |f(x)−f(y)|≥2, and if x and y are at distance two, then |f(x)−f(y)|≥1. The λ-number of G is the minimum span over all L(2,1)-labelings of G. Informally, an amalgamation of two graphs G1 and G2 along a fixed graph G0 is the simple graph obtained by identifying the vertices of two induced subgraphs isomorphic to G0, one in G1 and the other in G2. In this work, we supply a tight upper bound for the λ-number of amalgamations of several Cartesian products of complete graphs along a complete graph and find the exact λ-numbers for certain infinite subclasses of amalgamations of this form. A surprising relationship between the former upper bound and the minimum makespan scheduling problem is highlighted.

Integer invariants of abelian Cayley graphs

Let G be a finite abelian group, let E be a subset of G, and form the Cayley (directed) graph of G with connecting set E. We explain how, for various matrices associated to this graph, the spectrum can be used to give information on the Smith normal form. This technique is applied to several interesting examples, including matrices in the Bose–Mesner algebra of the Hamming association scheme H(n,q). We also recover results of Bai and Jacobson–Niedermaier–Reiner on the critical group of a Cartesian product of complete graphs.

On the Spanning Trees of the Hypercube and other Products of Graphs

We give two combinatorial proofs of a product formula for the number of spanning trees of the $n$-dimensional hypercube. The first proof is based on the assertion that if one chooses a uniformly random rooted spanning tree of the hypercube and orient each edge from parent to child, then the parallel edges of the hypercube get orientations which are independent of one another. This independence property actually holds in a more general context and has intriguing consequences. The second proof uses some "killing involutions'' in order to identify the factors in the product formula. It leads to an enumerative formula for the spanning trees of the $n$-dimensional hypercube augmented with diagonals edges, counted according to the number of edges of each type. We also discuss more general formulas, obtained using a matrix-tree approach, for the number of spanning trees of the Cartesian product of complete graphs.

Distinguishing numbers of Cartesian products of multiple complete graphs

We examine the distinguishing number of the Cartesian product of an arbitrary number of complete graphs. We show that for u 1 ≤ ... ≤ u d the distinguishing number of the Cartesian product of complete graphs of these sizes is either ⌈ u d 1/ s ⌉ or ⌈ u d 1/ s ⌉ + 1 where s = Π i = 1 d − 1 u i . In most cases, which of these values it is can be explicitly determined.

Geodesic pancyclicity and balanced pancyclicity of the generalized base-[formula omitted] hypercube

Recently, Chan et al. introduced geodesic-pancyclic graphs [H.C. Chan, J.M. Chang, Y.L. Wang, S.J. Horng, Geodesic-pancyclic graphs, Discrete Applied Mathematics 155 (15) (2007) 1971–1978] and weakly geodesic pancyclicity [H.C. Chan, J.M. Chang, Y.L. Wang, S.J. Horng, Geodesic-pancyclicity and fault-tolerant panconnectivity of augmented cubes, Applied Mathematics and Computation 207 (2009) 333–339]. Hsu et al. proposed a new cycle-embedding property called balanced pancyclicity [H.C. Hsu, P.L. Lai, C.H. Tsai, Geodesic pancyclicity and balanced pancyclicity of augmented cubes, Information Processing Letters 101 (2007) 227–232]. For a graph G(V,E) and any two vertices x and y of V, a cycle R containing x and y can be divided into two paths, Pt1 and Pt2, joining x and y such that len(Pt1)≤len(Pt2), where len(λ) denotes the length of the path λ. A geodesic cycle contains Pt1, which is the shortest path joining x and y in G, whereas, in a balanced cycle of an even (respectively, odd) length, len(Pt1)=len(Pt2) (respectively, len(Pt1)=len(Pt2)−1). A graph is weakly geodesic pancyclic (respectively, balanced pancyclic) if every two vertices x and y are contained in a geodesic cycle (respectively, balanced cycle) from Max(3,2Dist(x,y)) to N, where N is the order of the graph. The interconnection network considered in this paper is the generalized base-b hypercube, which is an attractive variant of the well-known hypercube. In fact, the generalized base-b hypercube is the Cartesian product of complete graphs with b vertices. The generalized base-b hypercube is superior to the hypercube in many criteria, such as diameter, connectivity, and fault diameter. In this paper, we study weakly geodesic pancyclicity and balanced pancyclicity of the generalized base-b hypercube. We show that the generalized base-b hypercube is weakly geodesic pancyclic for b≥3 and balanced pancyclic for b≥4.

Parallel construction of optimal independent spanning trees on Cartesian product of complete graphs

A set of k spanning trees rooted at the same vertex r in a graph G is said to be independent if for each vertex x other than r, the k paths from r to x, one path in each spanning tree, are internally disjoint. Using independent spanning trees (ISTs) one can design fault-tolerant broadcasting schemes and increase message security in a network. Thus, the problem of ISTs on graphs has been received much attention. Recently, Yang et al. proposed a parallel algorithm for generating optimal ISTs on the hypercube. In this paper, we propose a similar algorithm for generating optimal ISTs on Cartesian product of complete graphs. The algorithm can be easily implemented in parallel or distributed systems. Moreover, the proof of its correctness is simpler than that of Yang et al.

Formula omitted]-labelings of Hamming graphs

Given integers c ≥ 0 and h ≥ k ≥ 1 , a c - L ( h , k ) -labeling of a graph G is a mapping f : V ( G ) → { 0 , 1 , 2 , … , c } such that | f ( u ) − f ( v ) | ≥ h if d G ( u , v ) = 1 and | f ( u ) − f ( v ) | ≥ k if d G ( u , v ) = 2 . The L ( h , k ) -number λ h , k ( G ) of G is the minimum c such that G has a c - L ( h , k ) -labeling. The Hamming graph is the Cartesian product of complete graphs. In this paper, we study L ( h , k ) -labeling numbers of Hamming graphs. In particular, we determine λ h , k ( K n q ) for 2 ≤ q ≤ p with h / k ≤ n − q + 1 or 2 ≤ q ≤ p with h / k ≥ q n − 2 q + 2 or q = p + 1 with h / k ≤ n / p , where p is the minimum prime factor of n .

The distinguishing number of Cartesian products of complete graphs

The distinguishing number D(G) of a graph G is the least integer d such that G has a labeling with d labels that is preserved only by a trivial automorphism. We prove that Cartesian products of relatively prime graphs whose sizes do not differ too much can be distinguished with a small number of colors. We determine the distinguishing number of the Cartesian product Kk□Kn for all k and n, either explicitly or by a short recursion. We also introduce column-invariant sets of vectors and prove a switching lemma that plays a key role in the proofs.

Transitive decompositions of graph products: rank 3 grid type

A transitive decomposition is a pair (Γ, 𝒫) where Γ is a graph and 𝒫 is a partition of the arc set of Γ, such that there exists a group of automorphisms of Γ which leaves 𝒫 invariant and transitively permutes the parts in 𝒫. This paper concerns transitive decompositions where the group is a primitive rank 3 group of ‘grid’ type. The graphs Γ in this case are either products or Cartesian products of complete graphs. We first give some generic constructions for transitive decompositions of products and Cartesian products of copies of an arbitrary graph, and we then prove (except in a small number of cases) that all transitive decompositions with respect to a rank 3 group of grid type can be characterized using these constructions. Furthermore, the main results of this work provide a new proof and insight into the classification of rank 3 partial linear spaces of product action type studied by Devillers.

Constructing pairs of equienergetic and non-cospectral graphs

The energy of a simple graph G is the sum of the absolute values of the eigenvalues of its adjacency matrix. Two graphs of the same order are said to be equienergetic if they have the same energy. Several ways to construct equienergetic non-cospectral graphs of very large size can be found in the literature. The aim of this work is to construct equienergetic non-cospectral graphs of small size. In this way, we first construct several special families of such graphs, using the product and the cartesian product of complete graphs. Afterwards, we show how one can obtain new pairs of equienergetic non-cospectral graphs from the starting ones. More specifically, we characterize the connected graphs G for which the product and the cartesian product of G and K 2 are equienergetic non-cospectral graphs and we extend Balakrishnan’s result: For a non-trivial graph G , G ⊗ C 4 and G ⊗ K 2 ⊗ K 2 are equienergetic non-cospectral graphs, given in [R. Balakrishnan, The energy of a graph, Linear Algebra Appl. 387 (2004) 287–295].

Distinguishing colorings of Cartesian products of complete graphs

We determine the values of s and t for which there is a coloring of the edges of the complete bipartite graph K s , t which admits only the identity automorphism. In particular, this allows us to determine the distinguishing number of the Cartesian product of complete graphs.

Hamming polynomials and their partial derivatives

Hamming graphs are Cartesian products of complete graphs and partial Hamming graphs are their isometric subgraphs. The Hamming polynomial h ( G ) of a graph G is introduced as the Hamming subgraphs counting polynomial. K k -derivates ∂ k G ( k ≥ 2 ) of a partial Hamming graph are also introduced. It is proved that for a partial Hamming graph G , ∂ h ( G ) ∂ x k = h ( ∂ k G ) . A couple of combinatorial identities involving the coefficients of the Hamming polynomials of Hamming graphs are also proven.

Distance monotonicity and a new characterization of Hamming graphs

Hamming graphs are simply Cartesian products of complete graphs. Several characterizations of these graphs involve the notion of intervals in a graph and related topics. The notion of interval distance monotonicity has already been used in order to obtain a characterization of hypercubes. Here, we revisit this notion and use it to obtain a new characterization of Hamming graphs by performing a combination of the hypercube characterization and the interval hypercube characterization.

Factorizations of some weighted spanning tree enumerators

We give factorizations for weighted spanning tree enumerators of Cartesian products of complete graphs, keeping track of fine weights related to degree sequences and edge directions. Our methods combine Kirchhoff's Matrix-Tree Theorem with the technique of identification of factors.