Comb Graph Research Articles

Some Results on Odd-Even Congruence Labeling of Graphs

Introduction: Labeling of graphs has been introduced in 1966. Assignment of natural numbers to vertices and/or edges is referred as graph labeling. Inspired by the ample application of graph labeling technique in real life problems, multifarious labeling strategy was adopted and investigated by many researchers.
 Objectives:Graph labeling plays a vital role in various fields and can be implemented in multitudinous discipline including coding theory, X-ray, Psychology, crystallography, circuit design, communication networks, astronomy, radar, data security, secret sharing, data base management and so on. Apart from these labeling techniques serve as a model to understand discrete mathematical domains.
 Methodology:In this paper, an attempt has been made to introduce new labeling such as odd-even congruence labeling. Congruence Graph Labeling is an allocation of natural numbers as labels for the edges and vertices of a graph based on modular arithmetic property. Odd-even congruence labeling is an allocation of odd integers to vertices and even integers to edges in addition to congruence graph labeling.
 Result: The suggested labeling has been identified on complete bipartite graph, comb graph and spliting graph of a star graph. Further, it is proved that graph acquired by connecting two copies of even cycle Cr by a path Pt, K1,t⊗P2 and D2 (Pt) are odd-even congurence graph.

R-Chromatic Number On r-Dynamic Vertex Coloring of Comb Graph

Let be a graph with vertex set and edge set . An r-dynamic vertex coloring of a graph is a assigning colors to the vertices of such that for every vertex receives at least colors in its neighbors. The minimum color used in r-dynamic vertex coloring of graph is called the r-dynamic chromatic number denoted as . In this research we well determine the coloring pattern and the r-dynamic chromatic number of the comb graph , central graph of comb graph , middle graph of comb graph , line graph of comb graph , sub-division graf of comb graph , and para-line graph of comb graph Let be a graph with vertex set and edge set . An r-dynamic vertex coloring of a graph is a assigning colors to the vertices of such that for every vertex receives at least colors in its neighbors. The minimum color used in r-dynamic vertex coloring of graph is called the r-dynamic chromatic number denoted as . In this research we well determine the coloring pattern and the r-dynamic chromatic number of the comb graph , central graph of comb graph , middle graph of comb graph , line graph of comb graph , sub-division graf of comb graph , and para-line graph of comb graph

BILANGAN KETERHUBUNGAN TITIK PELANGI BEBERAPA KELAS GRAF

A graph G is called a rainbow vertex connected if every two vertices G are connected by a rainbow path, that is, a path whose all the internal vertices are of a different color. The rainbow vertex connection number of graph G denoted by rvc(G) is the minimum number of colors used to color all vertices by G such that the graph G is connected to rainbow vertex. The rainbow vertex connection number in a graph will not be less than the diameter of the graph minus one. The rainbow vertex connection number discussed in this article for various classes of graphs include complete graph Kn, complete bipartite graph Km,n , wheel graph Wn , two-layer wheel graph Wn2, complete multipartite graph Kn1,n2,...,nt , path Pn, comb graph GSn, graph , graph , graph , graph .
 Keywords: graph, vertex coloring, rainbow vertex connection number.

DIVISOR CORDIAL LABELING FOR SOME SNAKES AND DEGREE SPLITTING RELATED GRAPHS

For a graph G = (V (G),E(G)), the vertex labeling function is defined as a bijection f : V (G) → {1, 2, . . . , |V (G)|} such that an edge uv is assigned the label 1 if one f(u) or f(v) divides the other and 0 otherwise. f is called divisor cordial labeling of graph G if the number of edges labeled with 0 and the number of edges labeled with 1 differ by at most 1. In 2011, Varatharajan et al. [24] have introduced divisor cordial labeling as a variant of cordial labeling. In this paper, we study divisor cordial labeling for triangular snake and quadrilateral snake. Moreover, we investigate divisor cordial labeling for the degree splitting graph of path, shell, cycle with one chord, crown and comb graph.

On the smallest positive eigenvalue of bipartite unicyclic graphs with a unique perfect matching II

Let G be a simple graph with the adjacency matrix A ( G ) . Let τ ( G ) denote the smallest positive eigenvalue of A ( G ) . In 1990, Pavlíková and Kr c ˘ -Jediný proved that among all nonsingular trees on n = 2m vertices, the comb graph (obtained by taking a path on m vertices and adding a new pendant vertex to every vertex of the path) has the maximum τ value. We consider the problem for unicyclic graphs. Let U denote the class of all connected bipartite unicyclic graphs with a unique perfect matching, and for each m ≥ 3 , let U n be the subclass of U with graphs on n = 2m vertices. We first obtain the classes of unicyclic graphs U in U such that τ ( U ) ≤ 2 − 1 . We then find the unique graph U o n (resp. U e n ) having the maximum τ value among all graphs in U n when m is odd (resp. when m is even). Finally, we prove that U o 6 (the graph obtained from a cycle of order 4, by adding two pendants to two adjacent vertices) is the graph with maximum τ value among all graphs in U . As a consequence, we obtain a sharp upper bound for τ ( U ) when U ∈ U .

Clustering property for quantum Markov chains on the comb graph

<abstract><p>Quantum Markov chains (QMCs) on graphs and trees were investigated in connection with many important models arising from quantum statistical mechanics and quantum information. These quantum states generate many important properties such as quantum phase transition and clustering properties. In the present paper, we propose a construction of QMCs associated with an $ XX $-Ising model over the comb graph $ \mathbb N\rhd_0 \mathbb Z $. Mainly, we prove that the QMC associated with the disordered phase, enjoys a clustering property.</p></abstract>

Even Vertex <img src=image/13429275_01.gif>-Graceful Labeling on Rough Graph

The study of set of objects with imprecise knowledge and vague information is known as rough set theory. The diagrammatic representation of this type of information may be handled through graphs for better decision making. Tong He and K. Shi introduced the constructional processes of rough graph in 2006 followed by the notion of edge rough graph. They constructed rough graph through set approximations called upper and lower approximations. He et al developed the concept of weighted rough graph with weighted attributes. Labelling is the process of making the graph into a more sensible way. In this process, integers are assigned for vertices of a graph so that we will be getting distinct weights for edges. Weight of an edge brings the degree of relationship between vertices. In this paper we have considered the rough graph constructed through rough membership values and as well as envisaged a novel type of labeling called Even vertex <img src=image/13429275_02.gif>-graceful labeling as weight value for edges. In case of rough graph, weight of an edge will identify the consistent attribute even though the information system is imprecise. We have investigated this labeling for some special graphs like rough path graph, rough cycle graph, rough comb graph, rough ladder graph and rough star graph etc. This Even vertex <img src=image/13429275_02.gif>-graceful labeling will be useful in feature extraction process and it leads to graph mining.

On the smallest positive eigenvalue of bipartite unicyclic graphs with a unique perfect matching

Let G be a simple graph with the adjacency matrix A(G). Let τ(G) denote the smallest positive eigenvalue of A(G). In 1985, Godsil proved that in the class of all nonsingular trees on n=2m vertices, the path on 2m vertices has the minimum τ value. We consider the same problem for unicyclic graphs. Let Un be the class of connected bipartite unicyclic graphs on n=2m vertices, with a unique perfect matching. Let us take the path P2m=[1,2,…,m,m+1,…,2m]. If m is even, let Ue be the graph obtained from P2m by adding an edge between the vertices m−2 and m+3. If m is odd, let Uo be the graph obtained from the path P2m by adding an edge between the vertices m−3 and m+2. We show that the unique graph Ue (resp. Uo) has the minimum τ value among all the graphs in U2m when m is even (resp. when m is odd). In 1990, Pavlíková and Krč-Jediný proved that among all nonsingular trees on n=2m vertices the comb graph (obtained by taking a path on m vertices and adding a new pendant vertex to every vertex of the path) has the maximum τ value. Thus, it can be observed that there are only three trees with the smallest positive eigenvalue greater than 12. In this article, we characterize all bipartite unicyclic graphs G with a unique perfect matching such that τ(G)<12 (ultimately, we obtain only two bipartite unicyclic graphs with a unique perfect matching such that τ(G)>12).

Some Comb Related Cordial Graphs

In this research paper, we prove that different families of comb graph such as comb graph 𝐶𝑎𝑧, comb graph 𝐶𝑑𝑧, comb graph 𝐶𝑡𝑧, comb graph 𝐶𝑓𝑧 and comb graph 𝐶ℎ𝑧 are all cordial graphs.

On edge irregularity strength of line graph and line cut-vertex graph of comb graph

For a simple graph $G$, a vertex labeling $\phi:V(G) \rightarrow \{1, 2,\ldots,k\}$ is called $k$-labeling. The weight of an edge $xy$ in $G$, written $w_{\phi}(xy)$, is the sum of the labels of end vertices $x$ and $y$, i.e., $w_{\phi}(xy)=\phi(x)+\phi(y)$. A vertex $k$-labeling is defined to be an edge irregular $k$-labeling of the graph $G$ if for every two different edges $e$ and $f$, $w_{\phi}(e) \neq w_{\phi}(f)$. The minimum $k$ for which the graph $G$ has an edge irregular $k$-labeling is called the edge irregularity strength of $G$, written $es(G)$. In this paper, we find the exact value of edge irregularity strength of line graph of comb graph $P_n \bigodot K_1$ for $n=2,3,4$; and determine the bounds for $n \geq 5$. Also, the edge irregularity strength of line cut-vertex graph of $P_n \bigodot K_1$ for $n=2$; and determine the bounds for $n \geq 3$.

The Rainbow-Vertex Connection Number [RVCN] of Subdivision of Certain Graphs

Rainbow-Vertex Connection Number [rvcn] is computed for some graphs by the researchers. Here we have considered the subdivision graphs of certain graph classes. The rainbow edge connection number of subdivision of Triangular snake graph was already found[1]. Using the definition of rainbow-vertex connection number [5], which is the smallest positive integer k such that the graph is rainbow-vertex connected, we find the rainbow vertex number of subdivision graph of Friendship graph ,, Triangular snake graph and Comb graph .

Computing Y-index for some special graph

The topological index has transformed the chemical structure into a numeric number and they make a connection between chemistry and mathematics. In this paper, we compute Y-index for some special graphs in which it was named mk -graph and Jahangir graph. Also, we compute this topological index for subdivided mk -graph of path graph, comb graph, ladder graph, and triangular ladder graph. Finally, we calculate Y-index for another kind of special graph in which it was named Jahangir graph.

Strongly Multiplicative Labeling of Diamond Graph, Generalized Petersen Graph, and Some Other Graphs

A finite, simple graph of order k is said to be a strongly multiplicative graph when all vertices of the graph are labeled by positive integers 1,2,3, …, k such that the induced edge labels of the graph, obtained by the product of labels of end vertices of edges, are distinct. In this paper, we show that the diamond graph Brn for 𝑛 ≥ 3, umbrella graph Um,n, and generalized Petersen graph GP(n, k), for n ≥ 3 and 1 ≤ k &lt; (n/2), admit strongly multiplicative labeling. Moreover, strongly multiplicative labeling of a double comb graph and sunflower planar graph has also been investigated and elaborated as well with different examples.

Josephson Currents and Gap Enhancement in Graph Arrays of Superconductive Islands.

Evidence is reported that topological effects in graph-shaped arrays of superconducting islands can condition superconducting energy gap and transition temperature. The carriers giving rise to the new phase are couples of electrons (Cooper pairs) which, in the superconducting state, behave as predicted for bosons in our structures. The presented results have been obtained both on star and double comb-shaped arrays and the coupling between the islands is provided by Josephson junctions whose potential can be tuned by external magnetic field or temperature. Our peculiar technique for probing distribution on the islands is such that the hopping of bosons between the different islands occurs because their thermal energy is of the same order of the Josephson coupling energy between the islands. Both for star and double comb graph topologies the results are in qualitative and quantitative agreement with theoretical predictions.

Diffusive transport on networks with stochastic resetting to multiple nodes.

We study the diffusive transport of Markovian random walks on arbitrary networks with stochastic resetting to multiple nodes. We deduce analytical expressions for the stationary occupation probability and for the mean and global first passage times. This general approach allows us to characterize the effect of resetting on the capacity of random walk strategies to reach a particular target or to explore the network. Our formalism holds for ergodic random walks and can be implemented from the spectral properties of the random walk without resetting, providing a tool to analyze the efficiency of search strategies with resetting to multiple nodes. We apply the methods developed here to the dynamics with two reset nodes and derive analytical results for normal random walks and Lévy flights on rings. We also explore the effect of resetting to multiple nodes on a comb graph, Lévy flights that visit specific locations in a continuous space, and the Google random walk strategy on regular networks.

On (1, 2) - Domination of Certain Graphs

A vertex subset S of a graph G = (V,E) is called a (1,2)-dominating set if S is having the property that for every vertex v in V- S there is atleast one vertex in S of distance 1 from v and a vertex in S at a distance atmost 2 from v. The minimum cardinality of a (1, 2)-dominating set of G, denoted by ϒ (1, 2)(G), is called the (1, 2)-domination number of G. In this paper we discuss about the (1, 2)-dominating set of Shell graph C(n,n-3,), Jewel graph Jn and Comb graph Pn ʘ K1.

On r-dynamic coloring of comb graphs

An r-dynamic coloring of a graph G is a proper coloring of G such that every vertex in V(G) has neighbors in at least $\min\{d(v),r\}$ different color classes. The r-dynamic chromatic number of graph G denoted as $\chi_r (G)$, is the least k such that G has a coloring. In this paper we obtain the r-dynamic chromatic number of the central graph, middle graph, total graph, line graph, para-line graph and sub-division graph of the comb graph $P_n\odot K_1$ denoted by $C(P_n\odot K_1), M(P_n\odot K_1), T(P_n\odot K_1), L(P_n\odot K_1), P(P_n\odot K_1)$ and $S(P_n\odot K_1)$ respectively by finding the upper bound and lower bound for the r-dynamic chromatic number of the Comb graph.

Soft Graphs of Certain Graphs

Let G ∗ = (V, E) be a simple graph and A be any nonempty set of parameters. Let subset R of A×V be an arbitrary relation from A to V. A mapping from A to ᵖ(V) written as F:A → ᵖ(V) can be defined as F(x) = { y ∈ V/xRy} and a mapping from A to ᵖ(E) written as K:A → ᵖ(E) can be defined as K(x) = {uv ∈ E/{u, v} ⊆ F(x)}. The pair (F, A) is a soft set over V and the pair (K, A) is a soft set over E. Obviously (F(a), K(a)) is a subgraph of G ∗ for all a ∈ A. The 4-tuple G = ( G ∗, F, K, A) is called a soft graph of G. In this paper we discuss different soft graphs of graphs such as Complete graph, Star graph, Complete bipartite graph, Crown graph, Comb graph, Friendship graph, Bistar graph and Wheel graph.

Extra Skolem Difference Mean Labeling of Various Graphs

Let graph G=(V(G),E(G)) attains a Skolem difference mean labeling with p vertices and q edges is said to be an extra Skolem difference mean labeling of graph G if all the labels of the vertices are odd. The graph which attains an extra Skolem difference mean labeling is called an extra Skolem difference mean graph. We obtain an extra Skolem difference mean labeling for Comb graph, Twig of a path P_n, H graph of a path P_n, K_1,2*K_(1,n) graph, K_1,3*K_(1,n) graph, m- Join of H_n, P_n⊙K_(1,m) graph , HSS(P_n) graph, H⊙〖mK〗_1-graph of a path P_n.

Even odd Harmonious Labeling of Some Graphs

Let G = be a graph, with and . An injective mapping is called an even-odd harmonious labeling of the graph G, if an induced edge mapping such that (i) is bijective mapping (ii) The graph acquired from this labeling is called even-odd harmonious graph. In this paper, we discovered some interesting results like H-graph, comb graph, bistar graph and graph for even-odd harmonious labeling.