Comb Graphs Research Articles

Roman Domination in Mycielski Graphs: A Study of Some Graphs and a Heuristic Algorithm

Let G=(V,E) be a graph. A function f:V→\{0,1,2\}, if ∀u for which f(u)=0 is adjacent to ∃v for which f(v)=2, is called a Roman dominating function, and called in short terms RDF. The weight of an RDF f is f(V)=∑_(v∈V)▒f(v) . The Roman domination number of a graph G, denoted by γ_R (G), is the minimum weight of an RDF on G. This paper presents the results for Roman domination numbers of the Mycielski graphs obtained through Mycielski's construction of the comet, double comet, and comb graphs. An algorithm to determine the Roman domination number of any given graph is also provided.

An Edge Irregular Reflexive k−labeling of Comb Graphs with Additional 2 Pendants

Let G be a connected, simple, and undirrected graph, where V (G) is the vertex set and E(G) is the edge set. Let k be a natural numbers. For graph G we define a total k−labeling ρ such that the vertices of graph G are labeled with {0, 2, 4, . . . , 2kv} and the edges of graph G are labeled with {1, 2, 3, . . . , ke}, where k = max{2kv, ke}. Total k−labeling ρ called an edge irregular reflexive k− labeling if every two distinct edge of graph G have distinct edge weights, where the edge weight is defined as the sum of the label of that edge and the label of the vertices that are incident to this edge. The minimum k such that G has an edge irregular reflexive k−labeling called the reflexive edge strength of G. In this paper we determine the reflexive edge strength of some comb graphs.

An Edge Irregular Reflexive k−labeling of Comb Graphs with Additional 2 Pendants

Let G be a connected, simple, and undirrected graph, where V (G) is the vertex set and E(G) is the edge set. Let k be a natural numbers. For graph G we define a total k−labeling ρ such that the vertices of graph G are labeled with {0, 2, 4, . . . , 2kv} and the edges of graph G are labeled with {1, 2, 3, . . . , ke}, where k = max{2kv, ke}. Total k−labeling ρ called an edge irregular reflexive k− labeling if every two distinct edge of graph G have distinct edge weights, where the edge weight is defined as the sum of the label of that edge and the label of the vertices that are incident to this edge. The minimum k such that G has an edge irregular reflexive k−labeling called the reflexive edge strength of G. In this paper we determine the reflexive edge strength of some comb graphs.

A note on m-Zumkeller cordial labeling of graphs

Let G = (V,E) be a graph. An m-Zumkeller cordial labeling of the graph G is defined by an injective function f : V → N such that there exists an induced function f ∗ : E → {0, 1} defined by f ∗ (uv) = f (u) · f (v) that satisfies the following conditions- i. For every uv ∈ E, f ∗ (uv) = {█(1,if f (u) · f (v) is an m-Zumkeller number @ 0,otherwise )┤ ii. |ef ∗ (0) − ef∗ (1)| ≤ 1 where ef ∗ (0) and ef ∗ (1) denote the number of edges of the graph G having label 0 and 1 respectively under f ∗. In this paper we prove that there exist an m-Zumkeller cordial labeling of graphs in paths, cycles, comb graphs, ladder graphs, twig graphs, helm graphs, wheel graphs, crown graphs and star graphs.

Quantum Markov Chains on Comb Graphs: Ising Model

In the present paper, we construct quantum Markov chains (QMC) over the Comb graphs. As an application of this construction, it is proved the existence of the disordered phase for the Ising type models (within QMC scheme) over the Comb graphs. Moreover, it is also established that the associated QMC has clustering property with respect to translations of the graph. We stress that this paper is the first one where a nontrivial example of QMC over non-regular graphs is given.

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.

Quantum Markov Chains on Comb Graphs: Ising Model

Строятся квантовые цепи Маркова (КЦМ) над гребенчатыми графами. В качестве приложения этой конструкции доказано существование неупорядоченной фазы для моделей типа Изинга (в рамках схемы КЦМ) над гребенчатыми графами. Кроме того, установлено, что ассоциированная КЦМ обладает свойством кластеризации относительно сдвигов графа. Следует отметить, что настоящая работа является первой, где приводится нетривиальный пример КЦМ над нерегулярными графами.

On the zero forcing number and propagation time of oriented graphs

Zero forcing is a process of coloring in a graph in time steps known as propagation time. These graph-theoretic parameters have diverse applications in computer science, electrical engineering and mathematics itself. The problem of evaluating these parameters for a network is known to be NP-hard. Therefore, it is interesting to study these parameters for special families of networks. Perila et al. (2017) studied properties of these parameters for some basic oriented graph families such as cycles, stars and caterpillar networks. In this paper, we extend their study to more non-trivial structures such as oriented wheel graphs, fan graphs, friendship graphs, helm graphs and generalized comb graphs. We also investigate the change in propagation time when the orientation of one edge is flipped.

Spectra of comb graphs with tails

Given two graphs, a backbone and a finger, a comb product is a new graph obtained by grafting a copy of the finger into each vertex of the backbone. We study the comb graphs in the case when both components are the paths of order n and k, respectively, as well as the above comb graphs with an infinite ray attached to some of their vertices. A detailed spectral analysis is carried out in both situations.

K-Zumkeller Labeling of Graphs

In this paper, we mainly focus on to prove that the graphs, viz., (i)paths, (ii) comb graphs, (iii) cycles, (iv) ladder graphs and (v) Pn´Pn graphs are k-Zumkeller graphs.

BOSE–EINSTEIN CONDENSATION ON INHOMOGENEOUS AMENABLE GRAPHS

We investigate the Bose–Einstein condensation on nonhomogeneous amenable networks for the model describing arrays of Josephson junctions. The resulting topological model, whose Hamiltonian is the pure hopping one given by the opposite of the adjacency operator, also has a mathematical interest in itself. We show that for the nonhomogeneous networks like the comb graphs, particles condensate in momentum and configuration space as well. In this case different properties of the network, of geometric and probabilistic nature, such as the volume growth, the shape of the ground state, and the transience, all play a role in the condensation phenomena. The situation is quite different for homogeneous networks where just one of these parameters, e.g., the volume growth, is enough to determine the appearance of the condensation.

Asymptotics of the Euler number of bipartite graphs

We define the Euler number of a bipartite graph on n vertices to be the number of labelings of the vertices with 1 , … , n such that the vertices alternate in being local maxima and local minima. We reformulate the problem of computing the Euler number of certain subgraphs of the Cartesian product of a graph G with the path P m in terms of self-adjoint operators. The asymptotic expansion of the Euler number is given in terms of the eigenvalues of the associated operator. For two classes of graphs, the comb graphs and the Cartesian product P 2 □ P m , we numerically solve the eigenvalue problem.

COMB GRAPHS AND SPECTRAL DECIMATION

AbstractWe investigate the spectral properties of matrices associated with comb graphs. We show that the adjacency matrices and adjacency matrix Laplacians of the sequences of graphs show a spectral similarity relationship in the sense of work by L. Malozemov and A. Teplyaev (Self-similarity, operators and dynamics,Math. Phys. Anal. Geometry6(2003), 201–218), and hence these sequences of graphs show a spectral decimation property similar to that of the Laplacians of the Sierpiński gasket graph and other fractal graphs.

TOPOLOGY INDUCED MACROSCOPIC QUANTUM COHERENCE IN JOSEPHSON JUNCTION NETWORKS

We argue that Josephson junction networks may be engineered to allow for the emergence of new and robust quantum coherent states. We provide a rather intuitive argument showing how the change in topology may affect the quantum properties of a bosonic particle hopping on a network. As a paradigmatic example, we analyze in detail the quantum and thermodynamical properties of non-interacting bosons hopping on a comb graph. We show how to explicitly compute the inhomogeneities in the distribution of bosons along the comb's fingers, evidencing the effects of the topology induced spatial Bose–Einstein condensation characteristic of the system. We propose an experiment enabling to detect the spatial Bose–Einstein condensation for Josephson networks built on comb graphs.

Polyenes with maximum HOMO–LUMO gap

On the basis of a variable neighbourhood search with the AutoGraphiX software, it is conjectured that for even numbers of atoms the fully conjugated acyclic π system of maximum HOMO–LUMO gap is a `comb' in which each vertex of a backbone carries a single pendant edge. Chemically, this represents C n H 3 n/2+2 , an α, ω-diene with methylene groups attached at all intermediate positions. As n increases without limit, the conjectured asymptotic dimensionless gap is 2 2 −2 . Odd-numbered graphs that would have maximum gap as singly charged π systems are reported for up to 41 vertices; they are related to comb graphs.