Regular Graph Of Degree Research Articles

Regular graphs of degree at most four that allow two distinct eigenvalues

For an n×n matrix A, let q(A) be the number of distinct eigenvalues of A. If G is a connected graph on n vertices, let S(G) be the set of all real symmetric n×n matrices A=[aij] such that for i≠j, aij=0 if and only if {i,j} is not an edge of G. Let q(G)=min{q(A):A∈S(G)}. Studying q(G) has become a fundamental sub-problem of the inverse eigenvalue problem for graphs, and characterizing the case for which q(G)=2 has been especially difficult. This paper considers the problem of determining the regular graphs G that satisfy q(G)=2. The resolution is straightforward if the degree of regularity is 1,2, or 3. However, the 4-regular graphs with q(G)=2 are much more difficult to characterize. A connected 4-regular graph has q(G)=2 if and only if either G belongs to a specific infinite class of graphs, or else G is one of fifteen 4-regular graphs whose number of vertices ranges from 5 to 16. This technical result gives rise to several intriguing questions.

On Infinite, Cubic, Vertex-Transitive Graphs with Applications to Totally Disconnected, Locally Compact Groups

We study groups acting vertex-transitively and non-discretely on connected, cubic graphs (regular graphs of degree 3). Using ideas from Tutte's fundamental papers in 1947 and 1959, it is shown that if the action is edge-transitive, then the graph has to be a tree. When the action is not edge-transitive Tutte's ideas are still useful and can, amongst other things, be used to fully classify the possible two-ended graphs. Results about cubic graphs are then applied to Willis' scale function from the theory of totally disconnected, locally compact groups. Some of the results in this paper have most likely been known to experts but most of them are not stated explicitly with proofs in the literature.

On the vertex irregular reflexive labeling of several regular and regular-like graphs

A total k-labeling is defined as a function g from the edge set to the first natural number ke and a function f from the vertex set to a non-negative even number up to 2kv , where k = max{ke , 2kv }. A vertex irregular reflexive k-labeling of the graph G is total k-labeling if wt(x) ¹ wt(x¢) for every two different vertices x and x¢ of G, where wt(x) = f (x) + Σ xy ∈E(G) g(xy). The reflexive vertex strength of the graph G, denoted by rvs(G), is the minimum k for a graph G with a vertex irregular reflexive k-labeling. We will determine the exact value of rvs(G) in this paper, where G is a regular and regular-like graph. A regular graph is a graph where each vertex has the same number of neighbors. A regular graph with all vertices of degree r is called an r-regular graph or regular graph of degree r. A regular-like graphs is an almost regular graph that we develop in a new definition and we called it with (s, r) -almost regular graphs.

The spectrum on prism graph using circulant matrix

Spectral graph theory discusses about the algebraic properties of graphs based on the spectrum of a graph. This article investigated the spectrum of prism graph. The method used in this research is the circulant matrix. The results showed that prism graph P2,s is a regular graph of degree 3, for s odd and s ≥ 3, P2,s is a circulantt graph with regular spectrum.

Epidemic Source Detection in Contact Tracing Networks: Epidemic Centrality in Graphs and Message-Passing Algorithms

We study the epidemic source detection problem in contact tracing networks modeled as a graph-constrained maximum likelihood estimation problem using the susceptible-infected model in epidemiology. Based on a snapshot observation of the infection subgraph, we first study finite degree regular graphs and regular graphs with cycles separately, thereby establishing a mathematical equivalence in maximal likelihood ratio between the case of finite acyclic graphs and that of cyclic graphs. In particular, we show that the optimal solution of the maximum likelihood estimator can be refined to distances on graphs based on a novel statistical distance centrality that captures the optimality of the nonconvex problem. An efficient contact tracing algorithm is then proposed to solve the general case of finite degree-regular graphs with multiple cycles. Our performance evaluation on a variety of graphs shows that our algorithms outperform the existing state-of-the-art heuristics using contact tracing data from the SARS-CoV 2003 and COVID-19 pandemics by correctly identifying the superspreaders on some of the largest superspreading infection clusters in Singapore and Taiwan.

Tensor and Cartesian products for nanotori, nanotubes and zig–zag polyhex nanotubes and their applications to 13C NMR spectroscopy

We have developed novel topological techniques to generate distance degree vector sequences (DDS) and distance degree scalar sequences(SS) of tensor and Cartesian products of graphs. We establish that the tensor products of two distance degree regular graphs is distance degree regular. We apply the mathematical techniques thus developed for both tensor and Cartesian products to enumerate NMR signals and intensity patterns through the partitioning of equivalence classes accomplished by the use of distance degree vector sequences. In particular the techniques are applied to several nano materials such as carbon nanotori, monocyclic cylindrical nanotubes and zig–zag polyhex carbon nanotubes. It is shown that DDS sequences satisfactorily partition carbon vertices into equivalence classes for these nanomaterials. We also demonstrate applications of the developed techniques to diagrammatic representations of energy decompositions in molecular interactions.

Towards generalizing MacDougall’s conjecture on vertex-magic total labelings

MacDougall’s conjecture states that every regular graph of degree at least 2 has a vertex-magic total labeling (VMTL) with the lone exception of 2K3. Since there is enormous empirical evidence supporting this conjecture, it is reasonable to seek generalizations. Thus we ask the more general question: to what extent does the degree sequence of a graph determine the existence or nonexistence of a VMTL? We provide beginning steps towards answering this question, and related questions, by providing infinite families of degree sequences, and for each sequence, a graph with a VMTL and another graph without a VMTL.

Subsets of Cayley graphs that induce many edges

Let $G$ be a regular graph of degree $d$ and let $A\subset V(G)$. Say that $A$ is $\eta$-closed if the average degree of the subgraph induced by $A$ is at least $\eta d$. This says that if we choose a random vertex $x\in A$ and a random neighbour $y$ of $x$, then the probability that $y\in A$ is at least $\eta$. This paper was motivated by an attempt to obtain a qualitative description of closed subsets of the Cayley graph $\Gamma$ whose vertex set is $\mathbb{F}_2^{n_1}\otimes \dots \otimes \mathbb{F}_2^{n_d}$ with two vertices joined by an edge if their difference is of the form $u_1\otimes \cdots \otimes u_d$. For the matrix case (that is, when $d=2$), such a description was obtained by Khot, Minzer and Safra, a breakthrough that completed the proof of the 2-to-2 conjecture. In this paper, we formulate a conjecture for higher dimensions, and prove it in an important special case. Also, we identify a statement about $\eta$-closed sets in Cayley graphs on arbitrary finite Abelian groups that implies the conjecture and can be considered as a “highly asymmetric Balog--Szemerédi--Gowers theorem” when it holds. We conclude the paper by showing that this statement is not true for an arbitrary Cayley graph. It remains to decide whether the statement can be proved for the Cayley graph $\Gamma$.

Asymptotic normality of Laplacian coefficients of graphs

Let G be a simple graph with n vertices and letC(G;x)=∑k=0n(−1)n−kc(G,k)xk denote the Laplacian characteristic polynomial of G. Then if the size |E(G)| is large compared to the maximum degree Δ(G), Laplacian coefficients c(G,k) are approximately normally distributed (by central and local limit theorems). We show that Laplacian coefficients of the paths, the cycles, the stars, the wheels and regular graphs of degree d are approximately normally distributed respectively. We also point out that Laplacian coefficients of the complete graphs and the complete bipartite graphs are approximately Poisson distributed respectively.

Perfect quantum state transfer using Hadamard diagonalizable graphs

Quantum state transfer within a quantum computer can be achieved by using a network of qubits, and such a network can be modelled mathematically by a graph. Here, we focus on the corresponding Laplacian matrix, and those graphs for which the Laplacian can be diagonalized by a Hadamard matrix. We give a simple eigenvalue characterization for when such a graph has perfect state transfer at time π/2; this characterization allows one to choose the correct eigenvalues to build graphs having perfect state transfer. We characterize the graphs that are diagonalizable by the standard Hadamard matrix, showing a direct relationship to cubelike graphs. We then give a number of constructions producing a wide variety of new graphs that exhibit perfect state transfer, and we consider several corollaries in the settings of both weighted and unweighted graphs, as well as how our results relate to the notion of pretty good state transfer. Finally, we give an optimality result, showing that among regular graphs of degree at most 4, the hypercube is the sparsest Hadamard diagonalizable connected unweighted graph with perfect state transfer.

Antimagic Labeling of Regular Graphs

A graph $G=(V,E)$ is antimagic if there is a one-to-one correspondence $f: E \to \{1,2,\ldots, |E|\}$ such that for any two vertices $u,v$, $\sum_{e \in E(u)}f(e) \ne \sum_{e\in E(v)}f(e)$. It is known that bipartite regular graphs are antimagic and non-bipartite regular graphs of odd degree at least three are antimagic. Whether all non-bipartite regular graphs of even degree are antimagic remained an open problem. In this paper, we solve this problem and prove that all even degree regular graphs are antimagic. The paper was submitted to December 2014 by Journal of Graph Theory.

Circular chromatic indices of even degree regular graphs

The circular chromatic index of a graph G, written χc′(G), is the minimum r permitting a function c:E(G)→[0,r) such that 1≤|c(e)−c(e′)|≤r−1 whenever e and e′ are adjacent. It is known that if r∈(2+1k+1,2+1k) for some positive integer k, or r∈(113,4), then there is no graph G with χc′(G)=r. On the other hand, for any odd integer n≥3, if r∈[n,n+14], then there is a simple graph G with χc′(G)=r; if r∈[n,n+13], then there is a multigraph G with χc′(G)=r. For most reals r, it is unknown whether r is the circular chromatic index of a graph (or a multigraph) or not. In this paper, we prove that for any even integer n≥4, if r∈[n,n+1/6], then there is an n-regular simple graph G with χc′(G)=r; if r∈[n,n+1/3], then there is an n-regular multi-graph G with χc′(G)=r.

Distance Degree Regular Graphs and Distance Degree Injective Graphs: An Overview

The distance d(v,u) from a vertex v of G to a vertex u is the length of shortest v to u path. The eccentricity ev of v is the distance to a farthest vertex from v. If d(v,u) = e(v), (u ≠ v), we say that u is an eccentric vertex of v. The radius rad(G) is the minimum eccentricity of the vertices, whereas the diameter diam(G) is the maximum eccentricity. A vertex v is a central vertex if e(v) = rad(G), and a vertex is a peripheral vertex if e(v) = diam(G). A graph is self-centered if every vertex has the same eccentricity; that is, rad(G) = diam(G). The distance degree sequence (dds) of a vertex v in a graph G = (V, E) is a list of the number of vertices at distance 1, 2, ... . , e(v) in that order, where e(v) denotes the eccentricity of v in G. Thus, the sequence (di0,di1,di2, …, dij,…) is the distance degree sequence of the vertex vi in G where dij denotes the number of vertices at distance j from vi. The concept of distance degree regular (DDR) graphs was introduced by Bloom et al., as the graphs for which all vertices have the same distance degree sequence. By definition, a DDR graph must be a regular graph, but a regular graph may not be DDR. A graph is distance degree injective (DDI) graph if no two vertices have the same distance degree sequence. DDI graphs are highly irregular, in comparison with the DDR graphs. In this paper we present an exhaustive review of the two concepts of DDR and DDI graphs. The paper starts with an insight into all distance related sequences and their applications. All the related open problems are listed.

Classification of Knots of Small Complexity in Thickened Tori

We present a table of knots in a thickened torus T × I the diagrams of which have less than five crossing points. The knots are constructed by a three-step process: enumeration of regular graphs of degree 4, enumeration of all corresponding knot projections for each graph, and construction of minimal diagrams. The completeness of the table is proved.

Spanning Trees with many Leaves: New Lower Bounds in Terms of the Number of Vertices of Degree 3 and at Least 4

We prove, that every connected graph with $s$ vertices of degree 3 and $t$ vertices of degree at least~4 has a spanning tree with at least ${2\over 5}t +{1\over 5}s+\alpha$ leaves, where $\alpha \ge {8\over 5}$. Moreover, $\alpha \ge 2$ for all graphs besides three exclusions. All exclusion are regular graphs of degree~4, they are explicitly described in the paper. We present infinite series of graphs, containing only vertices of degrees~3 and~4, for which the maximal number of leaves in a spanning tree is equal for ${2\over 5}t +{1\over 5}s+2$. Therefore we prove that our bound is tight.

Extremal graphs for the geometric–arithmetic index with given minimum degree

Let G(k,n) be the set of connected simple n-vertex graphs with minimum vertex degree k. The geometric–arithmetic index GA(G) of a graph G is defined by GA(G)=∑uv2dudvdu+dv, where d(u) is the degree of vertex u and the summation extends over all edges uv of G. In this paper we find for k≥⌈k0⌉, with k0=q0(n−1), where q0≈0.088 is the unique positive root of the equation qq+q+3q−1=0, extremal graphs in G(k,n) for which the geometric–arithmetic index attains its minimum value, or we give a lower bound. We show that when k or n is even, the extremal graphs are regular graphs of degree k.

On automorphisms of strongly regular graphs with parameters (99, 42, 21, 15) and (99, 56, 28, 36)

We consider undirected graphs without loops or multiple edges. If a and b are vertices in a graph Γ, then d(a, b) denotes the distance between a and b, and Γi(a) denotes the subgraph of Γ induced by the set of vertices of Γ that are a distance of i away from a. The subgraph Γ1(a) is called the neighborhood of a and is denoted by [a]. By a ⊥ we denote the subgraph that is the ball of radius 1 centered at a. Γ is called a regular graph of degree k if [a] contains precisely k vertices for any vertex a in Γ. A graph Γ is – ≤

4-REGULAR GRAPH OF DIAMETER 2

A regular graph is a graph where each vertex has the same degree. A regular graph with vertices of degree k is called a k -regular graph or regular graph of degree k. Let G be a graph, the distance between two vertices in G is the number of edges in a shortest path connecting them. The diameter of G is the greatest distance between any pair of vertices. Let n4 be maximum integer number so that there exists an 4-regular graph on n4 vertices of diameter 2. We prove that n4 = 15.

Regular graphs with small second largest eigenvalue

We consider regular graphs with small second largest eigenvalue (denoted by ?2). In particular, we determine all triangle-free regular graphs with ?2 ? ?2, all bipartite regular graphs with ?2 ? ?3, and all bipartite regular graphs of degree 3 with ?2 ? 2.

Improved inapproximability results for counting independent sets in the hard‐core model

AbstractWe study the computational complexity of approximately counting the number of independent sets of a graph with maximum degree Δ. More generally, for an input graph and an activity , we are interested in the quantity defined as the sum over independent sets I weighted as . In statistical physics, is the partition function for the hard‐core model, which is an idealized model of a gas where the particles have non‐negligible size. Recently, an interesting phase transition was shown to occur for the complexity of approximating the partition function. Weitz showed an FPAS for the partition function for any graph of maximum degree Δ when Δ is constant and . The quantity is the critical point for the so‐called uniqueness threshold on the infinite, regular tree of degree Δ. On the other side, Sly proved that there does not exist efficient (randomized) approximation algorithms for , unless , for some function . We remove the upper bound in the assumptions of Sly's result for , that is, we show that there does not exist efficient randomized approximation algorithms for all for and . Sly's inapproximability result uses a clever reduction, combined with a second‐moment analysis of Mossel, Weitz and Wormald which prove torpid mixing of the Glauber dynamics for sampling from the associated Gibbs distribution on almost every regular graph of degree Δ for the same range of λ as in Sly's result. We extend Sly's result by improving upon the technical work of Mossel et al., via a more detailed analysis of independent sets in random regular graphs. © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 45, 78–110, 2014