Bridge Graphs Research Articles

Extension of bridge graphs and their chemical applications

Abstract A topological invariant is a numerical value which can be associated with a graph structure. These graph invariants are utilized for modeling information of molecules in structural chemistry and biology. Over the years many topological indices are proposed and studied based on degree, distance and other parameters of graph. Also graph invariants play a significant role in chemistry, especially, QSPR/QSAR studies. In this paper, we concentrate two graph invariants such as augmented Zagreb invariant and inverse sum indeg invariant of extended bridge graphs.

Multiplicative-sum Zagreb index of splice, bridge, and bridge-cycle graphs

The multiplicative-sum Zagreb index is a graph invariant defined as the product of the sums of the degrees of pairs of adjacent vertices in a graph. In this paper, we compute the multiplicative-sum Zagreb index of some composite graphs such as splice graphs, bridge graphs, and bridge-cycle graphs in terms of the multiplicative-sum Zagreb indices of their components. Then, we apply our results to compute the multiplicative-sum Zagreb index for several classes of chemical graphs and nanostructures.

On degree based topological indices of bridge graphs

In 2009, T. Mansour, M. Schork, introduce the concept of bridge graphs. A bridge graph B(G 1, G 2, … , Gm ) = B(G 1, G 2, … , Gm ; v 1, v 2, … , vm ) of with respect to the vertices is the graph obtained from the graphs G 1 ; G 2; … ; Gm by connecting the vertices vi and vi+1 by an edge for all I = 1; 2; … ; m-1. In this paper, we determine hyper-Zagreb index, first multiple Zagreb index, second multiple Zagreb index, Zagreb polynomials and M-polynomial for nano structures of bridge graphs.

The behavior of Tutte polynomials of graphs under five graph operations and its applications

The Tutte polynomial of a graph has many important applications in combinatorics, physics, and biology. Graph operations, such as triangulation and subdivision, have been widely used in building complex network models. In this paper, we show how the Tutte polynomial changes with five graph operations. Firstly, we study the connections between graph G and its operation graphs: the triangulation graph R(G), the diamond graph Z(G), the quadrilateral graph Q(G), the 2-triangulation graph R2(G), and the Wheatstone bridge graph W(G), in the respect of spanning subgraphs. Secondly, using these relations, we investigate the structure of the set of spanning subgraphs of each operation graph, and find that it is constituted by 2|E(G)| disjoint subsets. Then, we derive the contribution of each subset by an indirect method. Finally, we gain the Tutte polynomials of these five operation graphs. Moreover, we consider the Tutte polynomials of the pseudofractal scale-free network and a classic diamond hierarchical lattice as an application. Our technique can be applied to study other graph polynomials.

The impact of loops on the game of cops and robbers on graphs

In the game of cops and robbers on graphs, the cops and the robber are allowed to pass their turn if they are located on a looped vertex. This paper explores the effect of loops on the cop number and the capture time. We provide examples of graphs where the cop number almost doubles when the loops are removed, graphs where the cop number decreases when the loops are removed, graphs where the capture time is quadratic in the number of vertices and copwin graphs where the cop needs to move away from the robber in optimal play.

Hyperopic Cops and Robbers

We introduce a new variant of the game of Cops and Robbers played on graphs, where the robber is invisible unless outside the neighbor set of a cop. The hyperopic cop number is the corresponding analogue of the cop number, and we investigate bounds and other properties of this parameter. We characterize the cop-win graphs for this variant, along with graphs with the largest possible hyperopic cop number. We analyze the cases of graphs with diameter 2 or at least 3, focusing on when the hyperopic cop number is at most one greater than the cop number. We show that for planar graphs, as with the usual cop number, the hyperopic cop number is at most 3. The hyperopic cop number is considered for countable graphs, and it is shown that for connected chains of graphs, the hyperopic cop density can be any real number in [0,1/2].

Throttling for the game of Cops and Robbers on graphs

We consider the cop-throttling number of a graph G for the game of Cops and Robbers, which is defined to be the minimum of (k+captk(G)), where k is the number of cops and captk(G) is the minimum number of rounds needed for k cops to capture the robber on G over all possible games. We provide some tools for bounding the cop-throttling number, including showing that the positive semidefinite (PSD) throttling number, a variant of zero forcing throttling, is an upper bound for the cop-throttling number. We also characterize graphs having low cop-throttling number and investigate how large the cop-throttling number can be for a given graph. We consider trees, unicyclic graphs, incidence graphs of finite projective planes (a Meyniel extremal family of graphs), a family of cop-win graphs with maximum capture time, grids, and hypercubes. All the upper bounds on the cop-throttling number we obtain for families of graphs are O(n).

English

English

Max-min Rodeg index of bridge graphs and fullerenes

Chemical graph theory is a branch of mathematical chemistry which deals with the nontrivial applications of graph theory to solve molecular problems. A graph-based molecular descriptors commonly known as topological indices describe the structures of chemical compounds. They are used in the isomer discrimination, structure property relationship, structure activity relations. Also, the certain physicochemical properties such as boiling point, enthalpy of vaporization, stability, and so on are able to be predicted by this technique. Molecules and molecular compounds are often modeled by molecular graph which is a representation of the structural formula of a chemical compound in terms of graph theory. Recently, the Max-min rodeg index (Mm sde ) which is vertex degree-based topological index has attracted attention and gained popularity. This index give the best predictor for enthalpy of vaporization and standard enthalpy of vaporization in the set of octane isomers and also for log water activity coefficient in the set of polychlorobiphenyles. A fullerene graph is a cubic planar graph whose faces are pentagons and hexagons. In this study, the Max-min rodeg index of fullerenes and link of fullerenes is computed. Moreover, it is presented exact expressions for the Max-min rodeg index of bridge graphs.

The role of quantum correlations in Cop and Robber game

We introduce and study quantized versions of Cop and Robber game. We achieve this by using graph-preserving quantum operations, which are the quantum analogues of stochastic operations preserving the graph. We provide the tight bound for the number of operations required to reach the given state. By extending them to the controlled operations, we define a quantum-controlled Cop and Robber game, which expands the classical Cop and Robber game, as well as the classically controlled quantum Cop and Robber game. In contrast to the typical scheme for introducing quantum games, we assume that both parties can utilise full information about the opponent’s strategy. We show that the utilisation of the full knowledge about the opponent’s state does not provide the advantage. Moreover, the chances of catching the Robber decrease for classical cop-win graphs. This result does not depend on the chosen model of evolution. On the other hand, the possibility to execute controlled quantum operations allows catching the Robber on almost all classical cop-win graphs. By this, we demonstrate that it is necessary to enrich the structure of correlations between the players’ systems to provide a non-trivial quantized Cop and Robber game. Thus the quantum controlled operations offer a significant advantage over the classically controlled quantum operations.

3-colouring for dually chordal graphs and generalisations

In this paper, we investigate the Colourability problem for dually chordal graphs and some of its generalisations. We show that the problem remains NP-complete if limited to four colours. For the case of three colours, we present a simple linear time algorithm for dismantlable graphs (which include dually chordal graphs) and present an O(n3m) time algorithm for graphs with tree-breadth 1. Additionally, we show that a dually chordal graph is 3-colourable if and only if it is perfect and has no clique of size four.

Visibility graphs, dismantlability, and the cops and robbers game

We study versions of cop and robber pursuit–evasion games on the visibility graphs of polygons, and inside polygons with straight and curved sides. Each player has full information about the other player's location, players take turns, and the robber is captured when the cop arrives at the same point as the robber. In visibility graphs we show the cop can always win because visibility graphs are dismantlable, which is interesting as one of the few results relating visibility graphs to other known graph classes. We extend this to show that the cop wins games in which players move along straight line segments inside any polygon and, more generally, inside any simply connected planar region with a reasonable boundary. Essentially, our problem is a type of pursuit–evasion using the link metric rather than the Euclidean metric, and our result provides an interesting class of infinite cop-win graphs.

Zagreb Indices of Semi-Total(Total) Block Graphs of Bridge and Chain Graphs

Zagreb Indices of Semi-Total(Total) Block Graphs of Bridge and Chain Graphs

Cops and Robbers ordinals of cop-win trees

A relational characterization of cop-win graphs was provided by Nowakowski and Winkler in their seminal paper on the game of Cops and Robbers. As a by-product of that characterization, each cop-win graph is assigned a unique ordinal, which we refer to as a CR-ordinal. For finite graphs, CR-ordinals correspond to the length of the game assuming optimal play, with the cop beginning the game in a least favourable initial position. For infinite graphs, however, the possible values of CR-ordinals have not been considered in the literature until the present work.We classify the CR-ordinals of cop-win trees as either a finite ordinal, or those of the form α+ω, where α is a limit ordinal. For general infinite cop-win graphs, we provide an example whose CR-ordinal is not of this form. We finish with some problems on characterizing the CR-ordinals in the general case of cop-win graphs.

Bridged graphs, circuits and Fibonacci numbers

Series-parallel two-terminal graphs and corresponding unit resistor circuits are well explored. Here we expand the ideas to exclusive-bridged graphs and unit resistor circuits. We proof that both types of circuits have rational resistances whereof numerators and denominators of reduced fractions are smaller or equal to the Fibonacci number Fn+1. Series-parallel circuits satisfy another inequality. The sum of their numerator and denominator is smaller or equal to Fn+2. This is not true for exclusive-bridged circuits. The consequence is that combinations of these circuits or double-bridged circuits do not satisfy these inequalities. In a second part, counting series-parallel graphs and circuits is expanded to exclusive bridged graphs and circuits. This leads to new terms in two OEIS sequences.

Strong Spatial Mixing in Homomorphism Spaces

Given a countable graph $\mathcal{G}$ and a finite graph $H$, we consider $Hom(\mathcal{G}, H)$ the set of graph homomorphisms from $\mathcal{G}$ to $H$ and we study Gibbs measures supported on $Hom(\mathcal{G}, H)$. We develop some sufficient and other necessary conditions for the existence of Gibbs specifications on $Hom(\mathcal{G}, H)$ satisfying strong spatial mixing (with exponential decay rate). We relate this with previous work of Brightwell and Winkler, who showed that a graph $H$ has a combinatorial property called dismantlability iff for every $\mathcal{G}$ of bounded degree, there exists a Gibbs specification with unique Gibbs measure. We strengthen their result by showing that such Gibbs specification can be chosen to have weak spatial mixing. In addition, we exhibit a subfamily of graphs $H$ for which there exists Gibbs specifications satisfying strong spatial mixing, but we also show that there exist dismantlable graphs for which no Gibbs specification has strong spatial mixing.

F-Indices and its coindices of some classes of graphs

In this paper, first we investigate the basic properties of the F-index and its coindex of graph. Next we obtain the exact expression of F-indices and its coindices for bridge graph, chain graph and transformation of graph. Using some of these results, we have obtained the value of these indices for some important classes of chemical graphs.

On the geometric–arithmetic index by decompositions-CMMSE

The concept of geometric–arithmetic index was introduced in the chemical graph theory recently, but it has shown to be useful. There are many papers studying different kinds of indices (as Wiener, hyper–Wiener, detour, hyper–detour, Szeged, edge–Szeged, PI, vertex–PI and eccentric connectivity indices) under particular cases of decompositions. The main aim of this paper is to show that the computation of the geometric-arithmetic index of a graph G is essentially reduced to the computation of the geometric-arithmetic indices of the so-called primary subgraphs obtained by a general decomposition of G. Furthermore, using these results, we obtain formulas for the geometric-arithmetic indices of bridge graphs and other classes of graphs, like bouquet of graphs and circle graphs. These results are applied to the computation of the geometric-arithmetic index of Spiro chain of hexagons, polyphenylenes and polyethene.

The fifth geometric-arithmetic index of bridge graph and carbon nanocones

As an important topological index, the fifth geometric-arithmetic index is used to test the chemical properties of chemical compounds, nanomaterials and drugs. In this paper, by means of graph structure analysis and edge dividing technology, the formulate for calculating the fifth geometric-arithmetic index of bridge graph is given. Furthermore, we present the fifth geometric-arithmetic index of carbon nanocones .

Pursuit evasion on infinite graphs

The cop-and-robber game is a game between two players, where one tries to catch the other by moving along the edges of a graph. It is well known that on a finite graph the cop has a winning strategy if and only if the graph is constructible and that finiteness is necessary for this result.We propose the notion of weakly cop-win graphs, a winning criterion for infinite graphs which could lead to a generalisation. In fact, we generalise one half of the result, that is, we prove that every constructible graph is weakly cop-win. We also show that a similar notion studied by Chastand et al. (which they also dubbed weakly cop-win) is not sufficient to generalise the above result to infinite graphs.In the locally finite case we characterise the constructible graphs as the graphs for which the cop has a so-called protective strategy and prove that the existence of such a strategy implies constructibility even for non-locally finite graphs.