Butterfly Graph Research Articles

Exploration of CCD-Number for Some Specific Graphs

Objectives: A dominating set S of a graph G is said to be a complementary corona dominating set (CCD-set) if every vertex in is either a pendent vertex or a support vertex. The smallest cardinality of a CCD-set is called the CCD-number and is denoted by . In this article, we explore the CCD-number for some specific graphs. Methods: We are finding the upper bound of and the lower bound of . We prove that both the upper and lower bound are equal for the given graphs. Findings: In this article, we obtained the CCD-number for some specific graphs. Novelty: Complementary corona dominating set is a vertex set such that the induced subgraph of the complementary set is isomorphic to corona virus. Keywords: Dominating Set, Corona Dominating Set, Support And Pendent Vertex, Butterfly Graph, Dumbbell Graph, Coconut Tree Graph, Peacock Head Graph, Sunlet Graph, Spider Graph, Firecracker Graph, Uniform g-ply graph

Quantized fields induced topological features in Harper-Hofstadter model

Classical magnetic fields might change the properties of topological insulators such as the time reversal symmetry protected topological edge states. This poses a question that whether quantized fields would change differently the feature of topological materials with respect to the classical one. In this paper, we propose a model to describe topological insulators (ultracold atoms in square optical lattices with magnetic field) coupled to a tunable single-mode quantized field, and discuss the topological features of the system. We find that the quantized field can induce topological quantum phase transitions in a different way. To be specific, for fixed gauge magnetic flux ratio, we calculate the energy bands for different coupling constants between the systems and the fields in both open and periodic boundary conditions. We find that the Hofstadter butterfly graph is divided into a pair for continuous gauge magnetic flux ratio, which is different from the one without single-mode quantized field. In addition, we plot topological phase diagrams characterized by Chern number as a function of the momentum of the single-mode quantized field and obtain a quantized structure with non-zero filling factor.

A Ramsey Problem Related to Butterfly Graph vs. Small Paths and C3

A graph on five vertices consisting of 2 copies of the cycle graph C3 sharing a common vertex is called the Butterfly graph (B). The smallest natural number s such that any two-colouring (say red and blue) of the edges of Kj×s has a copy of a red B or a blue G is called the multipartite Ramsey number of Butterfly graph versus G. This number is denoted by mj(B,G). In this paper, we find the exact values for mj(B,G) when j >2 and G represents any small path or else three cycles.

Research on electromagnetic characteristics of metal spheres with variable parameter defects based on COMSOL finite element method

In order to study the influence of defect geometric parameters (width, thickness) on the electromagnetic characteristics of the space around the defect in the process of eddy current testing of metal spheres, based on the principle of eddy current testing, a metal sphere detection model with different defect parameters was established by COMSOL software for simulation. The distribution of the magnetic field around the defect under different parameters is analyzed with the help of the butterfly diagram composed of the radial magnetic induction intensity and the axial magnetic induction intensity. The results show that: When the width of the corrosion defect is different, the butterfly diagram will change differently. When the width is small, the butterfly diagram basically does not change. When the width reaches a certain value, the butterfly diagram begins to change significantly. The intersection of the graph and the initial butterfly graph is defined as the influence node, and the width of the defect can be quantitatively analyzed according to the sequence of the influence nodes. The influence of the thickness of the corrosion defect on the magnetic field on the surface of the sphere is different under different defect widths. The larger the thickness of the corrosion defect, the greater the influence on the magnetic field near the center of the defect, and the deeper the depression of the butterfly diagram.

Geometry, number theory, and the butterfly spectrum of two-dimensional Bloch electrons.

We take a deeper dive into the geometry and the number theory that underlay the butterfly graphs of the Harper and the generalized Harper models of Bloch electrons in a magnetic field. The root of the number theoretical characteristics of the fractal spectrum is traced to a close relationship between the Farey tree-the hierarchical tree that generates all rationals and the Wannier diagram-a graph that labels all the gaps of the butterfly graph. The resulting Farey-Wannier hierarchical lattice of trapezoids provides a geometrical representation of the nested pattern of butterflies in the butterfly graph. Some features of the energy spectrum, such as absence of some of the Wannier trajectories in the butterfly graph falling outside the number theoretical framework, can be stated as a simple rule of minimal violation of mirror symmetry. In a generalized Harper model, number theoretical framework prevails with the Farey-Wannier hierarchical lattice regrouping to form some hexagonal cells creating different species of butterflies.

What do Bloch electrons in a magnetic field have to do with Apollonian packing of circles?

Integral Apollonian packing, the packing of circles with integer curvatures, where every circle is tangent to three other mutually tangent circles, is shown to encode the fractal structure of the energy spectrum of two-dimensional Bloch electrons in a magnetic field, known as the ‘Hofstadter butterfly’. In this Apollonian–butterfly-connection, the integer curvatures of the circles contain in a convoluted form, the topological quantum numbers of the butterfly graph—the quanta of the Hall conductivity. Nesting properties of these two fractals are described in terms of the Apollonian group and the conformal transformations. In this mapping, Farey tree hierarchy plays the central role, revealing how the geometry and the number theory are intertwined in the quantum mechanics of Bloch electrons in a magnetic field.

Power of 2 Decomposition of a Complete Tripartite Graph K2,4,M and a Special Butterfly Graph

Let G be a finite, connected simple graph with p vertices and q edges. If G1 , G2 ,…, Gn are connected edge-disjoint subgraphs of G with E(G) = E(G1 )  E(G2 )  …  E(Gn) , then {G1 , G2 , …, Gn} is said to be a decomposition of G. A graph G is said to have Power of 2 Decomposition if G can be decomposed into edge-disjoint subgraphs G G G n  2 4 2 , ,..., such that each G i 2 is connected and ( ) 2 , i E Gi  for 1  i  n. Clearly, 2[2 1] n q . In this paper, we investigate the necessary and sufficient condition for a complete tripartite graph K2,4,m and a Special Butterfly graph           3 2 5 BF 2m 1 to accept Power of 2 Decomposition.

Edge δ− Graceful Labeling for Some Cyclic-Related Graphs

In this paper, we introduce a new type of labeling of a graph G with p vertices and q edges called edge δ− graceful labeling, for any positive integer δ, as a bijective mapping f of the edge set EG into the set δ,2δ,3δ,⋯,qδ such that the induced mapping f∗:VG→0,δ,2δ,3δ,⋯,qδ−δ, given by f∗u=∑uv∈EGfuvmodδk, where k=maxp,q, is an injective function. We prove the existence of an edge δ− graceful labeling, for any positive integer δ, for some cycle-related graphs like the wheel graph, alternate triangular cycle, double wheel graph Wn,n, the prism graph Πn, the prism of the wheel PWn, the gear graph Gn, the closed helm CHn, the butterfly graph Bn, and the friendship Frn.

On Prime Cordial Labeling of Crown, Armed Crown, H-graph and Butterfly graph

The prime cordial labeling for crown graph, armed crown, H-graph and butterfly graph have been described in this paper.

Interlace polynomials of friendship graphs

In this paper, we study the interlace polynomials of friendship graphs, that is, graphs that satisfy the Friendship Theorem given by Erdos, Renyi and Sos. Explicit formulas, special values and behavior of coefficients of these polynomials are provided. We also give the interlace polynomials of other similar graphs, such as, the butterfly graph.

A Study on Total Coloring and Finding the Total Chromatic Number of Cycle, Butterfly and Web Graph

A Study on Total Coloring and Finding the Total Chromatic Number of Cycle, Butterfly and Web Graph

On the strong metric dimension of generalized butterfly graph, starbarbell graph, and graph

Let G be a connected graph with vertex set V(G) and edge set E(G). For every pair of vertices , the interval I[u, v] between u and v to be the collection of all vertices that belong to some shortest u − v path. A vertex strongly resolves two vertices u and v if u belongs to a shortest v − s path or v belongs to a shortest u − s path. A vertex set S of G is a strong resolving set of G if every two distinct vertices of G are strongly resolved by some vertex of S. The strong metric basis of G is a strong resolving set with minimal cardinality. The strong metric dimension sdim(G) of a graph G is defined as the cardinality of strong metric basis. In this paper we determine the strong metric dimension of a generalized butterfly graph, starbarbell graph, and graph. We obtain the strong metric dimension of generalized butterfly graph is sdim(BFn) = 2n − 2. The strong metric dimension of starbarbell graph is . The strong metric dimension of graph are for m > 3 and n = 2, and for m > 3 and n > 2.

On the Total Edge Irregularity Strength of Generalized Butterfly Graph

Let G(V, E) be a connected, simple, and undirected graph with vertex set V and edge set E. A total k-labeling is a map that carries vertices and edges of a graph G into a set of positive integer labels {1, 2, …, k}. An edge irregular total k-labeling λ: V(G) ∪ E(G) → {1, 2, …, k} of a graph G is a total k-labeling such that the weights calculated for all edges are distinct. The weight of an edge uv in G, denoted by wt(uv), is defined as the sum of the label of u, the label of v, and the label of uv. The total edge irregularity strength of G, denoted by tes(G), is the minimum value of the largest label k over all such edge irregular total k-labelings. A generalized butterfly graph, BFn, obtained by inserting vertices to every wing with assumption that sum of inserting vertices to every wing are same then it has 2n + 1 vertices and 4n − 2 edges. In this paper, we investigate the total edge irregularity strength of generalized butterfly graph, BFn, for n > 2. The result is .

STRENGTH OF CERTAIN FUZZY GRAPHS

In this paper we find the strength of a strong fuzzy wheel graph, strong fuzzy complete bipartite graph, strong fuzzy butterfly graph, strong fuzzy bull graph and also determined the strength of a properly linked fuzzy graph with complete fuzzy graphs as its parts. AMS Subject Classification: 05C72

Butterfly Graphs with Shell Orders m and 2m+1 are Graceful

A graceful labelling of an un directed graph G with n edges is a one-one function from the set of vertices V(G) to the set {0, 1, ,2, . . ., n} such that the induced edge labels are all distinct. An induced edge label is the absolute difference between the two end vertex labels. A shell graph is defined as a cycle Cn with (n - 3) chords sharing a common end point called the apex . A double shell is one vertex union of two shells. A bow graph is defined to be a double shell in which each shell has any order. In this paper we define a butterfly graph as a bow graph with exactly two pendant edges at the apex and we prove that all butterfly graphs with one shell of order m and the other shell of order (2m + 1) are graceful.

Pairwise Intersession Network Coding on Directed Networks

When there exists only a single multicast session in a directed acyclic/cyclic network, the existence of a network coding solution is characterized by the classic min-cut/max-flow theorem. For the case of more than one coexisting sessions, network coding also demonstrates throughput improvement over noncoded solutions. This paper proposes pairwise intersession network coding, which allows for arbitrary directed networks but restricts the coding operations to being between two symbols (for acyclic networks) or between two strings of symbols (for cyclic networks). A graph-theoretic characterization of pairwise intersession network coding is proven based on paths with controlled edge-overlap. This new characterization generalizes the edge-disjoint path characterization of noncoded network communication and includes the well-studied butterfly graph as a special case. Based on this new characterization, various aspects of pairwise intersession network coding are studied, including the sufficiency of linear codes, the complexity of identifying coding opportunities, its topological analysis, and bandwidth- and coding-efficiency.

Generalized Butterfly Graph and Its Application to Video Stream Authentication

This paper presents the generalized butterfly graph (GBG) and its application to video stream authentication. Compared with the original butterfly graph, the proposed GBG provides significantly increased flexibility, which is necessary for streaming applications, including supporting arbitrary bit-rate budget for authentication and arbitrary number of video packets. Within the GBG design, the problem of constructing an authentication graph is defined as follows: given the total number of packets to protect, the expected packet loss rate for the network, and the available overhead budget, how should one design the authentication graph to maximize the probability that the received packets are verifiable? Furthermore, given the fact that media packets are typically of unequal importance, we explore two variants of the GBG authentication, packet sorting and unequal authentication protection, which apply unequal treatment to different packets based on their importance. Lastly, we examine how the proposed GBG authentication can be applied within the context of rate-distortion-authentication (R-D-A) optimized streaming: given a media stream protected by GBG authentication, the R-D-A optimized streaming technique computes an optimized transmission schedule by recognizing and accounting for the authentication dependencies in the GBG authentication graph.

The forwarding indices of wrapped butterfly networks

AbstractLet G be a connected graph. A routing in G is a set of fixed paths for all ordered pairs of vertices in G. The forwarding index of G is the minimum of the largest number of paths specified by a routing passing through any vertex of G taken over all routings in G. This article investigates the forwarding index of a wrapped butterfly graph, determines the exact value for the directed case, and gives an upper bound for undirected case. © 2008 Wiley Periodicals, Inc. NETWORKS, 2009

Cycle embedding in faulty wrapped butterfly graphs

AbstractIn this paper, we study the maximal length of cycle embedding in a faulty wrapped butterfly graph BFn with at most two faults in vertices and/or edges. When there is one vertex fault and one edge fault, we prove that the maximum cycle length is n2n − 2 if n is even and n2n − 1 if n is odd. When there are two faulty vertices, the maximum cycle length is n2n − 2 for odd n. All these results are optimal because the wrapped butterfly graph is bipartite if and only if n is even. © 2003 Wiley Periodicals, Inc.

Hamilton Cycle Decomposition of the Butterfly Network

In this paper, we prove that the wrapped Butterfly graph WBF(d,n) of degree d and dimension n is decomposable into Hamilton cycles. This answers a conjecture of Barth and Raspaud who solved the case d = 2.