Regular Complete Bipartite Graph Research Articles

Permanents of almost regular complete bipartite graphs

Let G be a graph, and let A ( G ) be the adjacency matrix of G. The computation of permanent of A ( G ) is #p-complete. Computing permanent of A ( G ) is of great interest in quantum chemistry, statistical physics, among other disciplines. In this paper, we characterize the ordering of permanents of adjacency matrices of all graphs obtained from regular complete bipartite graph K p , p by deleting six edges. As an application, we show that all graphs with a perfect matching obtained from K p , p with six edges deleted are determined by their permanental polynomials.

Generalized polygons and star graphs of cyclic presentations of groups

Groups defined by presentations for which the components of the corresponding star graph are the incidence graphs of generalized polygons are of interest as they are small cancellation groups that – via results of Edjvet and Vdovina – are fundamental groups of polyhedra with the generalized polygons as links and so act on Euclidean or hyperbolic buildings; in the hyperbolic case the groups are SQ-universal. A cyclic presentation of a group is a presentation with an equal number of generators and relators that admits a particular cyclic symmetry. We obtain a classification of the concise cyclic presentations where the components of the corresponding star graph are generalized polygons. The classification reveals that both connected and disconnected star graphs are possible and that only generalized triangles (i.e. incidence graphs of projective planes) and regular complete bipartite graphs arise as the components. We list the presentations that arise in the Euclidean case and show that at most two of the corresponding groups are not SQ-universal (one of which is not SQ-universal, the other is unresolved). We obtain results that show that many of the SQ-universal groups are large.

Eulerian orientations and vertex-connectivity

It is well-known that every Eulerian orientation of an Eulerian 2k-edge-connected undirected graph is k-arc-connected. A long-standing goal in the area has been to obtain analogous results for vertex-connectivity. Levit, Chandran and Cheriyan recently proved in Levit et al. (2018) that every Eulerian orientation of a hypercube of dimension 2k is k-vertex-connected. Here we provide an elementary proof for this result.We also show other families of 2k-regular graphs for which every Eulerian orientation is k-vertex-connected, namely the even regular complete bipartite graphs, the incidence graphs of projective planes of odd order, the line graphs of regular complete bipartite graphs and the line graphs of complete graphs.Furthermore, we provide a simple graph counterexample for a conjecture of Frank attempting to characterize graphs admitting at least one k-vertex-connected orientation.

Search on vertex-transitive graphs by lackadaisical quantum walk

The lackadaisical quantum walk is a discrete-time, coined quantum walk on a graph with a weighted self-loop at each vertex. It uses a generalized Grover coin and the flip-flop shift, which makes it equivalent to Szegedy's quantum Markov chain. It has been shown that a lackadaisical quantum walk can improve spatial search on the complete graph, discrete torus, cycle, and regular complete bipartite graph. In this paper, we observe that these are all vertex-transitive graphs, and when there is a unique marked vertex, the optimal weight of the self-loop equals the degree of the loopless graph divided by the total number of vertices. We propose that this holds for all vertex-transitive graphs with a unique marked vertex. We present a number of numerical simulations supporting this hypothesis, including search on periodic cubic lattices of arbitrary dimension, strongly regular graphs, Johnson graphs, and the hypercube.

Quantum walk search on the complete bipartite graph

The coined quantum walk is a discretization of the Dirac equation of relativistic quantum mechanics, and it is the basis of many quantum algorithms. We investigate how it searches the complete bipartite graph of $N$ vertices for one of $k$ marked vertices with different initial states. We prove intriguing dependence on the number of marked and unmarked vertices in each partite set. For example, when the graph is irregular and the initial state is the typical uniform superposition over the vertices, then the success probability can vary greatly from one timestep to the next, even alternating between 0 and 1, so the precise time at which measurement occurs is crucial. When the initial state is a uniform superposition over the edges, however, the success probability evolves smoothly. As another example, if the complete bipartite graph is regular, then the two initial states are equivalent. Then if two marked vertices are in the same partite set, the success probability reaches 1/2, but if they are in different partite sets, it instead reaches $1$. This differs from the complete graph, which is the quantum walk formulation of Grover's algorithm, where the success probability with two marked vertices is 8/9. This reveals a contrast to the continuous-time quantum walk, whose evolution is governed by Schr\"odinger's equation, which asymptotically searches the regular complete bipartite graph with any arrangement of marked vertices in the same manner as the complete graph.

On 2-distance-transitive circulants

A complete classification is given of 2-distance-transitive circulants, which shows that a 2-distance-transitive circulant is a cycle, a Paley graph of prime order, a regular complete multipartite graph, or a regular complete bipartite graph of order twice an odd integer minus a 1-factor.

Robust cycle bases do not exist for [formula omitted] if [formula omitted

A basis for the cycle space of a graph is said to be robust if any cycle Z of G is a sum Z=C1+C2+⋯+Ck of basis elements such that (i) (C1+C2+⋯+Cℓ−1)∩Cℓ is a nontrivial path for each 2≤ℓ<k. Hence, (ii) each partial sum C1+C2+⋯+Cℓ is a cycle for 1≤ℓ≤k. While complete graphs and 2-connected plane graphs have robust cycle bases, it is shown that regular complete bipartite graphs Kn,n do not have any robust cycle basis if n≥8.

On the Cost of Waking Up

Often, in a distributed system, a task must be performed in which all entities must be involved; however only some of them are active, while the others are inactive, unaware of the new computation that has to take place. In these situations, all entities must become active, a task known as Wake-Up. It is not difficult to see that Broadcast is just the special case of the Wake-Up problem, when there is only one initially active entity. Both problems can be solved with the same trivial but expensive solution: Flooding. More efficient broadcast protocols exist for some classes of dense interconnection networks. The research question we examine is whether also wake-up can be performed significantly better in three classes of regular interconnection networks: hypercubes, complete networks, and regular complete bipartite graphs. In a d -dimensional hypercube network of n nodes, the cost of broadcasting is Theta( n ) even if the edge labeling is arbitrary and the network is asynchronous. We show that, instead, wake-up requires Omega( n log n ) message transmissions in the worst case, even if the network is synchronous and has sense of direction. Similarly, in a regular complete bipartite network K p,p of n =2 p anonymous entities the cost of broadcasting is Theta( n ) even if the edge labeling is arbitrary and the network is asynchronous; instead, we show that wake-up requires Theta( n 2 ) message transmissions in the worst case, even if the network is synchronous and has sense of direction. In a complete network K n of n entities, the cost of broadcasting is minimal: n -1 message transmissions suffice even if the entities are anonymous. In this paper we prove that the cost of wake-up is order of magnitude higher. In the case of anonymous entities, Omega( n 2 ) message transmissions are needed in the worst case, even if the network is fully synchronous and has sense of direction. In the case of entities with distinct ids, Omega( n log n ) transmissions need to be performed and the bound is tight. This shows that, when the entities have Ids, Wake-Up is computationally as costly as the apparently more complex Election problem.

On the characterizing properties of the permanental polynomials of graphs

Let G be a graph and A(G) the adjacency matrix of G. The permanental polynomial of G is defined as π(G,x)≜per(xI-A(G)). If two graphs G and H have the same permanental polynomial, then G is called a per-cospectral mate of H. A graph G is said to be characterized by its permanental polynomial if all the per-cospectral mates of G are isomorphic to G. It is shown that complete graphs, stars, regular complete bipartite graphs, and odd cycles are characterized by their permanental polynomials. We prove that in general the permanental polynomial cannot characterize the paths and even cycles. In particular, for each l⩾1 and m⩾2, we can find non-isomorphic per-cospectral mates of P4l+3 and C4m, respectively. When we restrict our consideration to connected graphs, both the paths and even cycles C4l+2 are characterized by their permanental polynomials.

Kirchhoff index in line, subdivision and total graphs of a regular graph

Let G be a connected regular graph and l(G), s(G), t(G) the line, subdivision, total graphs of G, respectively. In this paper, we derive formulae and lower bounds of the Kirchhoff index of l(G), s(G) and t(G), respectively. In particular, we give special formulae for the Kirchhoff index of l(G), s(G) and t(G), where G is a complete graph Kn, a regular complete bipartite graph Kn,n and a cycle Cn.

Distinguishing labellings of group action on vector spaces and graphs

Suppose Γ is a group acting on a set X. A k-labeling of X is a mapping c : X → { 1 , 2 , … , k } . A labeling c of X is distinguishing (with respect to the action of Γ) if for any g ∈ Γ , g ≠ id X , there exists an element x ∈ X such that c ( x ) ≠ c ( g ( x ) ) . The distinguishing number, D Γ ( X ) , of the action of Γ on X is the minimum k for which there is a k-labeling which is distinguishing. This paper studies the distinguishing number of the linear group GL n ( K ) over a field K acting on the vector space K n and the distinguishing number of the automorphism group Aut ( G ) of a graph G acting on V ( G ) . The latter is called the distinguishing number of the graph G and is denoted by D ( G ) . We determine the value of D GL n ( K ) ( K n ) for all fields K and integers n. For the distinguishing number of graphs, we study the possible value of the distinguishing number of a graph in terms of its automorphism group, its maximum degree, and other structure properties. It is proved that if Aut ( G ) = S n and each orbit of Aut ( G ) has size less than ( n 2 ) , then D ( G ) = ⌈ n 1 / k ⌉ for some positive integer k. A Brooks type theorem for the distinguishing number is obtained: for any graph G, D ( G ) ⩽ Δ ( G ) , unless G is a complete graph, regular complete bipartite graph, or C 5 . We introduce the notion of uniquely distinguishable graphs and study the distinguishing number of disconnected graphs.

A decreasing sequence of upper bounds on the largest Laplacian eigenvalue of a graph

Let G be a simple graph. In this paper, we obtain a sequence ( b p ) p=1 ∞ of upper bounds on the largest eigenvalue λ 1( G) of the Laplacian matrix of G. Then, we show that this sequence converges to λ 1( G) and that ( b 2 p ) p=0 ∞ is a monotone strictly decreasing sequence except if G is a complete graph or G is a star graph or G is a regular complete bipartite graph. For these graphs, b p = λ 1( G) for all p. The bounds b 1 and b 2 are discussed.

Extending cycles in graphs

A cycle C in a graph G is extendable if there exists a cycle C' in G such that V(C)⊆V(C') and |V(C')| = |V(C)| + 1. A graph G is cycle extendable if G has at least one cycle and every nonhamiltonian cycle is extendable. A graph G of order p⩾3 has a pancyclic ordering if its vertices can be labelled v1, v2,…, vp so that the subgraph of G induced by v1, v2,…, vk contains a cycle of length k, for each k∈ {3, 4,…, p};.The theme of this paper is to investigate to what extent known sufficient conditions for a graph to be hamiltonian imply the extendability of cycles. A number of theorems and conjectures are stated. For example, it is shown that if C is a nonextendable cycle in a graph satisfying Ore's sufficient condition for a hamiltonian cycle then the subgraph induced by the vertices of C is either a complete graph or a regular complete bipartite graph. Results are also given relating to extremal problems, stability, graphs with forbidden induced subgraphs, squares of graphs and chordal graphs.

On randomly 3-axial graphs

A graph G, every vertex of which has degree at least three, is randomly 3-axial if for each vertex v of G, any ordered collection of three paths in G of length one with initial vertex v can be cyclically randomly extended to produce three internally disjoint paths which contain all the vertices of G. Randomly 3-axial graphs of order p &gt; 4 are characterized for p ≢ 1 (mod 3), and are shown to be either complete graphs or certain regular complete bipartite graphs.