Windmill Graph Research Articles

On vertex and edge eccentricity-based topological indices of a certain chemical graph that represents bidentate ligands

Graph theory is the most powerful tools in the mathematics and computer science, also study of descriptors in quantitative structure property relationship (QSPR) and quantitative structure activity relationship (QSAR) studies in the chemistry science. Let G=(V(G),E(G)) be a simple molecular graph without directed and multiple edges and without loops. A topological index is a numerical descriptor of the molecular structure derived from the corresponding molecular graph. There are a lot of topological indices for molecular graphs. Eccentricity-based topological indices such as vertex (-edge) eccentric and modified vertex (-edge) eccentric connectivity indices are very important QSPR/QSAR studies. In this paper, the edge eccentric and modified vertex (-edge) eccentric connectivity indices are computed for the Dutch windmill graph Dmn that represents bidentate ligands. Also, we plotted the two-dimensional graphics of the Dmn with the help of cartesian coordinate system.

Embedding Complete Bipartite Graphs into Necklace Graphs

Abstract Graph embedding is an important technique used in studying the problem of efficiently implementing parallel algorithms on parallel computers. Wirelength is an embedding parameter widely studied in data structures and data representations, electrical networks, VLSI network and chemical graphs. This parameter had been studied for embedding complete bipartite graphs into complete necklace, star necklace and windmill graphs.

Distance and eccentricity based invariants of windmill graph

In the fields of chemical graph theory, molecular topology, and mathematical chemistry, a topological index also known as a connectivity index is a type of a molecular descriptor that is calculated based on the molecular graph of a chemical compound. In this paper, we computed the Hosoya polynomial, Wiener index and hyper Wiener index of Windmill graph. We also computed Harary polynomial, Schultz polynomial, modified Schultz polynomial,eccentric connectivity polynomial, generalized Harary index, multiplicative Wiener index, Schultz index, modified Schultz index, eccentric connectivity index, etc for understudy families of graphs.

The graph tessellation cover number: Chromatic bounds, efficient algorithms and hardness

A tessellation of a graph is a partition of its vertices into vertex disjoint cliques. A tessellation cover of a graph is a set of tessellations that covers all of its edges, and the tessellation cover number is the size of the smallest tessellation cover. These concepts are motivated by their application to quantum walk models, in special, the evolution operator of the staggered model is obtained from a graph tessellation cover. We show that the minimum between the chromatic index of the graph and the chromatic number of its clique graph, which we call chromatic upper bound, is tight with respect to the tessellation cover number for star-octahedral and windmill graphs; whereas for (3,p)-extended wheel graphs, the tessellation cover number is 3 and the chromatic upper bound is 3p. The t-tessellability problem aims to decide whether there is a tessellation cover of the graph with t tessellations. Using graph classes whose tessellation cover numbers achieve the chromatic upper bound, we obtain that t-tessellability is polynomial-time solvable for bipartite, {triangle, proper major}-free, threshold, and diamond-free K-perfect graphs; whereas is NP-complete for triangle-free for t≥3, unichord-free for t≥3, planar for t=3, biplanar for t≥3, chordal (2,1)-graphs for t≥4, (1,2)-graphs for t≥4, and diamond-free with diameter at most five for t=3. We improve the complexity of 2-tessellability problem to linear time.

Hamming Index for Several Graph Classes Produced by an Side-Node Incidence Matrix

For example G(V,E) becomes graph on p nodes and q sides and B’(G) called being side-node incidence matrix from G. Every side e of G can be labeled using a binary digit string of length n from the row of B’(G) which corresponds to the side e, denoted by s(e). The Hamming distance between the strings s(e) and s(f) of length n is defined to be the number of positions of s(e) and s(f) with different digit. Hamming index from a graph is the number of Hamming distances between all pairs of strings. In this research, author discuss the Hamming index of graphs produced by an side-node incidence matrix, particularly windmill graph and snake graphs.

Wirelength of Enhanced Hypercube into Windmill and Necklace Graphs

An embedding of an interconnection network into another is one of the main issues in parallel processing and computing systems. Congestion, dilation, expansion and wirelength are some of the parameters used to analyze the efficiency of an embedding in which resolving the wirelength problem reduces time and cost in the embedded design. Due to the potential topological properties of enhanced hypercube, it has become constructive in recent years, and a lot of research work has been carried out on it. In this paper, we use the edge isoperimetric problem to produce the exact wirelengths of embedding enhanced hypercube into windmill and necklace graphs.

Eccentricity-Based Topological Invariants of Some Chemical Graphs

Topological index is an invariant of molecular graphs which correlates the structure with different physical and chemical invariants of the compound like boiling point, chemical reactivity, stability, Kovat’s constant etc. Eccentricity-based topological indices, like eccentric connectivity index, connective eccentric index, first Zagreb eccentricity index, and second Zagreb eccentricity index were analyzed and computed for families of Dutch windmill graphs and circulant graphs.

New Results of Face Labeling for Some Plane Graphs

A labeling of a plane graph is called super d-antimagic if the vertices receive the smallest labels and the weight set of all faces in an arithematic progression with difference d, where weight of each face is the some of all labels correspond to that face. In this paper, first we construct an upper bound for the paprmeter d. Secondly, we show that if a plane graph G is super d-antimagic then its subdivision is also super d-antimagic. Finally, we show that the Dutch windmill graph exist super d-antimagic labeling for different values of d, s.

Optimal information ratio of secret sharing schemes on Dutch windmill graphs

One of the basic problems in secret sharing is to determine the exact values of the information ratio of the access structures. This task is important from the practical point of view, since the security of any system degrades as the amount of secret information increases. A Dutch windmill graph consists of the edge-disjoint cycles such that all of them meet in one vertex. In this paper, we determine the exact information ratio of secret sharing schemes on the Dutch windmill graphs. Furthermore, we determine the exact ratio of some related graph families.

On generalized windmill graphs

In this paper we will consider generalizations of the class of so-called windmill graphs, which were recently introduced by Estrada [6]. A windmill graph W(η,k) consists of η copies of the complete graph Kk, with every node connected to a common node. Estrada [6] showed that the clustering coefficient and the transitivity index of windmill graphs diverge, when the graph size tends to infinity. In addition [6] studied the spectra of the adjacency and the Laplacian matrices of these graphs. In this paper we will generalize the family of windmill graphs in three ways. We will study properties for all three types of generalized windmill graphs. In particular we will focus on the behavior of the two clustering metrics. We will quantify the difference between the two metrics, under various conditions. In addition, we give the spectra of the adjacency and the Laplacian matrices of these graphs. We will also derive analytic expressions for several other graph metrics, such as average path length, heterogeneity index and a variety of robustness metrics. We also show how the generalized windmill graphs can be used to construct pairs of non-isomorphic graphs with the same number of nodes and links. Finally, we will show how generalized windmill graphs occur quite naturally in the study of public transportation networks.

G-Convex Weight Sequences

In this paper, we introduce the notion of g-convex weight sequence (gcws) for connected graphs based on the concept of g-convexity and g-weight. g-weight is a natural generalization of the notion of branch weight for trees. We investigate the various questions of realization of an integer sequence as a g-convex weight sequence for trees and some special classes of graphs such as complete graphs, windmill and degenerate windmill graphs and wheels.

3-total edge product cordial labeling of some new classes of graphs

In this paper we study 3-total edge product cordial (3-TEPC) labeling of some new classes of graphs. We study ladder graph, wheel graph, m isomorphic copies of n–cycle graph and dutch windmill graph in the context of 3-TEPC labeling. We also studied 3-TEPC labeling of some particular examples of graph operations like corona graphs and subdivision graphs.

On the strong metric dimension of sun graph, windmill graph, and möbius ladder graph

Let G be a connected graph with vertex set V(G) and edge set E(G). The interval I[u, v] between u and v is defined as 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 set is a strong resolving set if every two distinct vertices of G are strongly resolved by some vertex of S. The smallest cardinality of strong resolving set is called a strong metric basis. The strong metric dimension of G, denoted by sdim(G), is defined as the cardinality of the strong metric basis. In this paper we determine the strong metric dimension of a sun graph Sn, a windmill graph , and a Möbius ladder graph Mn. We obtain the strong metric dimension of sun graph Sn is n - 1 for n > 3. The strong metric dimension of windmill graph is (n - 1)m - 1 for m > 2 and n > 3. The strong metric dimension of Möbius ladder graph Mn with n > 5 is for n even.

Computation of Multiplicative Topological Indices of Dutch Windmill Graph

Computation of Multiplicative Topological Indices of Dutch Windmill Graph

The Odd Harmonious Labeling on Variation of the Double Quadrilateral Windmill Graphs

A Graphis calledif it hasvertices and edqes. The graphis said to be an odd harmonious if there exists an injectionsuch that induced mappingis a bijection and is said to be an odd harmonious labeling of. In this paper, we construct the odd harmonious labeling on variation of the double quadrilateral windmill graph with . Moreover, we also give odd harmonious labeling construction for generalized variation of the double quadrilateral windmill graph withand . Keywords: Odd harmonious labeling, variation, double quadrilteral, windmill graph.

Odd Harmonious Labeling on Pleated of the Dutch Windmill Graphs

A graph G(p,q) with p=|V(G)| vertices and q=|E(G)| edges. The graph G(p,q) is said to be odd harmonious if there exist an injection f: V(G)-{0,1,2,...,2q-1} such that the induced function f*: E(G)-{1,2,3,...,2q-1} defined by f*(uv)=f(u)+f(v) which is a bijection and f is said to be odd harmonious labeling of G(p,q). In this paper we prove that pleated of the Dutch windmill graphs C_4^(k)(r) with k=1 and r=1 are odd harmonious graph. Moreover, we also give odd harmonious labeling construction for the union pleated of the Dutch windmill graph C_4^(k)(r) union C_4^(k)(r) with k=1 and r=1.

On the chromatic number of (P_{5},windmill)-free graphs

In this paper we study the chromatic number of \((P_5, windmill)\)-free graphs. For integers \(r,p\geq 2\) the windmill graph \(W_{r+1}^p=K_1 \vee pK_r\) is the graph obtained by joining a single vertex (the center) to the vertices of \(p\) disjoint copies of a complete graph \(K_r\). Our main result is that every \((P_5, windmill)\)-free graph \(G\) admits a polynomial \(\chi\)-binding function. Moreover, we will present polynomial \(\chi\)-binding functions for several other subclasses of \(P_5\)-free graphs.

Computation of Topological Indices of Dutch Windmill Graph

In this paper, we compute Atom-bond connectivity index, Fourth atom-bond connectivity index, Sum connectivity index, Randic connectivity index, Geometric-arithmetic connectivity index and Fifth geometric-arithmetic connectivity index of Dutch windmill graph.

On the Laplacian energy of Windmill graph and graph D_{m,C_n}

On the Laplacian energy of Windmill graph and graph D_{m,C_n}

When local and global clustering of networks diverge

The average Watts–Strogatz clustering coefficient and the network transitivity are widely used descriptors for characterizing the transitivity of relations in real-world graphs (networks). These indices are bounded between zero and one, with low values indicating poor transitivity and large ones indicating a high proportion of closed triads in the graphs. Here, we prove that these two indices diverge for windmill graphs when the number of nodes tends to infinity. We also give evidence that this divergence occurs in many real-world networks, especially in citation and collaboration graphs. We obtain analytic expressions for the eigenvalues and eigenvectors of the adjacency and the Laplacian matrices of the windmill graphs. Using this information we show the main characteristics of two dynamical processes when taking place on windmill graphs: synchronization and epidemic spreading. Finally, we show that many of the structural and dynamical properties of a real-world citation network are well reproduced by the appropriate windmill graph, showing the potential of these graphs as models for certain classes of real-world networks.