Bridge Graphs Research Articles

Complete characterization of bridge graphs with local antimagic chromatic number 2

An edge labeling of a connected graph $G = (V, E)$ is said to be local antimagic if it is a bijection $f\colon E \to\{1,\ldots ,|E|\}$ such that for any pair of adjacent vertices $x$ and $y$, $f^+(x)\not= f^+(y)$, where the induced vertex label $f^+(x)= \sum f(e)$, with $e$ ranging over all the edges incident to $x$. The local antimagic chromatic number of $G$, denoted by $\chi_{la}(G)$, is the minimum number of distinct induced vertex labels over all local antimagic labelings of $G$. In this paper, we characterize $s$-bridge graphs with local antimagic chromatic number 2.

Quantum Max-flow in the Bridge Graph

Quantum Max-flow in the Bridge Graph

On cop-win dynamic graphs

We extend the Nowakowski–Winkler characterisation of cop-win graphs to dynamic graphs and we extend the Hahn–MacGillivray algorithm for recognising k-cop-win graphs to dynamic graphs.

First-order logic axiomatization of metric graph theory

The main goal of this note is to provide a First-Order Logic with Betweenness (FOLB) axiomatization of the main classes of graphs occurring in Metric Graph Theory, in analogy to Tarski's axiomatization of Euclidean geometry. We provide such an axiomatization for weakly modular graphs and their principal subclasses (median and modular graphs, bridged graphs, Helly graphs, dual polar graphs, etc), basis graphs of matroids and even Δ-matroids, partial cubes and their subclasses (ample partial cubes, tope graphs of oriented matroids and complexes of oriented matroids, bipartite Pasch and Peano graphs, cellular and hypercellular partial cubes, almost-median graphs, netlike partial cubes), and Gromov hyperbolic graphs. On the other hand, we show that some classes of graphs (including chordal, planar, Eulerian, and dismantlable graphs), closely related with Metric Graph Theory, but defined in a combinatorial or topological way, do not allow such an axiomatization.

Bridge Graph Attention based Graph Convolution Network with Multi-Scale Transformer for EEG Emotion Recognition

Bridge Graph Attention based Graph Convolution Network with Multi-Scale Transformer for EEG Emotion Recognition

A cop-winning strategy on strongly cop-win graphs

Nowakowski and Winkler [5] introduced a game on graphs where a cop tries to catch a robber; in particular they showed that the cop has a winning strategy on a (finite or infinite) graph if and only if the graph is dismantlable. In this paper, we prove a similar result for a game on a finite graph where the robber can travel through arbitrary many edges in a single move. The original dismantling procedure consists in deleting specific vertices one by one until reaching a single vertex. This procedure fails with a fast robber because by deleting vertices we remove possible escape ways for the robber, and hence possibly change the nature of the graph from robber-win to cop-win. To overcome this problem, we define a new dedicated light-dismantling procedure, which is a generalization of the classical one, with a broader application possibility, not only to the fast robber game. Instead of being deleted, vertices are darkened: dark vertices can be only visited by the robber, in passing. Hence we introduce undergrounded graphs, on which the game will be played. We define a monotonic winning strategy for the cop that ensures that she is progressing towards the robber, thus securing a growing area.

Topological Evaluation of Certain Computer Networks by Contraharmonic-Quadratic Indices

In various fields, different networks are used, most of the time not of a single kind; but rather a mix of at least two networks. These kinds of networks are called bridge networks which are utilized in interconnection networks of PC, portable networks, spine of internet, networks engaged with advanced mechanics, power generation interconnection, bio-informatics and substance intensify structures. Any number that can be entirely calculated by a graph is called graph invariants. Countless mathematical graph invariants have been portrayed and utilized for connection investigation during the latest twenty years. Nevertheless, no trustworthy evaluation has been embraced to pick, how much these invariants are associated with a network graph or sub-atomic graph. In this paper, it will discuss three unmistakable varieties of bridge networks with an incredible capacity of assumption in the field of computer science, chemistry, physics, drug industry, informatics and arithmetic in setting with physical and manufactured developments and networks, since Contraharmonic-quadratic invariants (CQIs) are recently presented and have different figure qualities for different varieties of bridge graphs or networks. The study settled the geography of bridge graphs/networks of three novel sorts with two kinds of CQI and Quadratic-Contraharmonic Indices (QCIs). The deduced results can be used for the modeling of the above-mentioned networks.

Analysis of Bridge Graph by Means of K-Banhatti Sombor Invariants

Any number that can be uniquely determined by a graph is called a graph invariant. During the last twenty years’ countless mathematical graph invariants have been characterized and utilized for correlation analysis. However, no reliable examination has been embraced to decide, how much these invariants are related with a network graph or molecular graph. In this paper, it will discuss three different variants of bridge networks with good potential of prediction in the field of computer science, mathematics, chemistry, pharmacy, informatics and biology in context with physical and chemical structures and networks, because K-Banhatti Sombor invariants are freshly presented and have numerous prediction qualities for different variants of bridge graphs or networks. These deduced results can be used for the modelling of computer networks like LAN, MAN, WAN, backbone of internet and other networks/structures of computers, power generation, bio-informatics and chemical compound.

On a contraction-like property of dismantlable graphs

In a recent paper, Baradol, Gopal and Radenović (2020) study the contraction properties of mappings on rectangular metric spaces endowed with a digraph. In this paper, we show that a class of mappings on dismantlable undirected graphs have a natural contraction-like property acting on a spanning tree of the graph. We propose further characterizations of dismantlable graphs and apply our results to pursuit-evasion games on graphs.

Discovering Irregularities from Computer Networks by Topological Mapping

Any number that can be uniquely identified and varied by a graph is known as a graph invariant. This paper will talk about three unique variations of bridge networks, sierpinski networks, honeycomb, and hexagonal networks, with great capability of forecast in the field of software engineering, arithmetic, physics, drug store, informatics, and chemistry in setting with physical and chemical properties. Irregularity sombor invariant is newly introduced and has various expectation characteristics for various variations of bridge graphs or other networks, as mentioned. First, find the irregularities in the networks with the help of the Irregularity sombor index. This will be performed in a step by step procedure. The study will take an existing network, associate it with a graph after finding their vertices and edges, then solve the topology of a graph of a network. Graphical results demonstrate the upper and lower bounds and irregularities of certain networks, and mathematical results are used for modeling purposes. The review settled the topologies of graphs/networks of seven distinct sorts with an Irregularity sombor index. These concluded outcomes can be utilized for the demonstration and modeling of computer networks such as local area networks, Metropolitan area networks, Wide area networks, memory interconnection networks, processor interconnection networks, the spine of the internet, and different networks/designs of Personal computers, power generation networks, mobile base station and chemical compound amalgamation and so on.

On Some Properties of Non-traceable Cubic Bridge Graph

Graphs considered in this paper are simple finite undirected graphs without loops or multiple edges. A simple graphs where each vertex has degree 3 is called cubic graphs. A cubic graphs, that is, 1-connected or cubic bridge graph is traceable if its contains Hamiltonian path otherwise, non-traceable. In this paper, we introduce a new family of cubic graphs called Non-Traceable Cubic Bridge Graph (NTCBG) that satisfy the conjecture of Zoeram and Yaqubi (2017). In addition, we defined two important connected component of NTCBG that is, central fragment that give assurance for a graph to be non-traceable and its branches. Some properties of NTCBG such as chromatic number and clique number were also provided.

Constructible graphs and pursuit

A (finite or infinite) graph is called constructible if it may be obtained recursively from the one-point graph by repeatedly adding dominated vertices. In the finite case, the constructible graphs are precisely the cop-win graphs, but for infinite graphs the situation is not well understood.One of our aims in this paper is to give a graph that is cop-win but not constructible. This is the first known such example. We also show that every countable ordinal arises as the rank of some constructible graph, answering a question of Evron, Solomon and Stahl. In addition, we give a finite constructible graph for which there is no construction order whose associated domination map is a homomorphism, answering a question of Chastand, Laviolette and Polat.Lehner showed that every constructible graph is a weak cop win (meaning that the cop can eventually force the robber out of any finite set). Our other main aim is to investigate how this notion relates to the notion of ‘locally constructible’ (every finite graph is contained in a finite constructible subgraph). We show that, under mild extra conditions, every locally constructible graph is a weak cop win. But we also give an example to show that, in general, a locally constructible graph need not be a weak cop win. Surprisingly, this graph may even be chosen to be locally finite. We also give some open problems.

DIMENSI METRIK GRAF HASIL OPERASI JEMBATAN DARI CATERPILLAR HOMOGEN DAN POT BUNGA DIPERUMUM

The metric dimension is a concept that has many applications, such as robotic navigation. This concept will distinguish each vertex of a graph based on some vertices. The distinguishing vertices are called the basis of the graph. Let G be a connected graph, the metric dimension, dim(G), is the smallest cardinality of the basis of graph G. On this paper, we present the metric dimensions of the bridge graph of a homogeneous caterpillar graph Cm,n and a generalized flower pot graph $C_p-K_{(q_1, q_2,\cdos,q_p}$. This research was conducted by the approach of structure analysis by location of the bridge vertices, the edge of the bridge, and the order of the graph. The results show that the metric dimensions of the bridge graph are at least can be reduced at most 2, and the maximum values are the same as the value of $m(n-1)+ \sum_{i=1}^p q_i- 2p$.Keywords: Metric dimension, caterpillar, unicyclic, bridgeMSC2020: 05C12

On Generalized Topological Indices for Some Special Graphs

Topological indices are numeric values associated with a graph and characterize its structure. There are various topological indices in graph theory such as degree‐based, distance‐based, and counting‐related topological indices. Among these indices, degree‐based indices are very interesting and studied well in literature. In this work, we studied the generalized form of harmonic, geometric‐arithmetic, Kulli–Basava indices, and generalized power‐sum‐connectivity index for special graph that are bridge graph over path, bridge graph over cycle, bridge graph over complete graph, wheel graph, gear graph, helm graph, and square lattice graph. We found exact values for the stated indices and for the stated special graphs. We also investigated the generalized form of the indices for various properties of alkane isomers, from which we obtained interesting results which are closed to that of experimental obtained results.

Constructible Graphs And Pursuit

A (finite or infinite) graph is called constructible if it may be obtained recursively from the one-point graph by repeatedly adding dominated vertices. In the finite case, the constructible graphs are precisely the cop-win graphs, but for infinite graphs the situation is not well understood. One of our aims in this paper is to give a graph that is cop-win but not constructible. This is the first known such example. We also show that every countable ordinal arises as the rank of some constructible graph, answering a question of Evron, Solomon and Stahl. In addition, we give a finite constructible graph for which there is no construction order whose associated domination map is a homomorphism, answering a question of Chastand, Laviolette and Polat. Lehner showed that every constructible graph is a weak cop win (meaning that the cop can eventually force the robber out of any finite set). Our other main aim is to investigate how this notion relates to the notion of `locally constructible' (every finite graph is contained in a finite constructible subgraph). We show that, under mild extra conditions, every locally constructible graph is a weak cop win. But we also give an example to show that, in general, a locally constructible graph need not be a weak cop win. Surprisingly, this graph may even be chosen to be locally finite. We also give some open problems.

K-Banhatti Sombor Invariants of Certain Computer Networks

Any number that can be uniquely determined by a graph is called a graph invariant. During the last twenty years’ countless mathematical graph invariants have been characterized and utilized for correlation analysis. However, no reliable examination has been embraced to decide, how much these invariants are related with a network graph or molecular graph. In this paper, it will discuss three different variants of bridge networks with good potential of prediction in the field of computer science, mathematics, chemistry, pharmacy, informatics and biology in context with physical and chemical structures and networks, because k-banhatti sombor invariants are freshly presented and have numerous prediction qualities for different variants of bridge graphs or networks. The study solved the topology of a bridge graph/networks of three different types with two invariants K-Banhatti Sombor Indices and its reduced form. These deduced results can be used for the modeling of computer networks like Local area network (LAN), Metropolitan area network (MAN), and Wide area network (WAN), backbone of internet and other networks/structures of computers, power generation, bio-informatics and chemical compounds synthesis.

K-Banhatti Invariants Empowered Topological Investigation of Bridge Networks

Any number that can be uniquely determined by a graph is called graph invariants. During the most recent twenty years’ innumerable numerical graph invariants have been described and used for correlation analysis. In the fast and advanced environment of manufacturing of networks and other products which used different networks, no dependable assessment has been embraced to choose, how much these invariants are connected with a network graph or molecular graph. In this paper, it will talk about three distinct variations of bridge networks with great capability of expectation in the field of computer science, chemistry, physics, drug industry, informatics, and mathematics in setting with physical and synthetic constructions and networks, since K-Banhatti invariants are newly introduced and have various forecast characteristics for various variations of bridge graphs or networks. The review settled the topology of bridge graph/networks of three unique sorts with three types of K-Banhatti Indices. These concluded outcomes can be utilized for the modeling of interconnection networks of Personal computers (PC), networks like Local area network (LAN), Metropolitan area network (MAN) and Wide area network (WAN), the spine of internet and different networks/designs of PCs, power generation interconnection, bio-informatics and chemical structures.

Cops and Robber on some families of oriented graphs

Cops and Robber is a two-player pursuit-evasion game in which a set of cops, controlled by Player 1, try to capture the robber, controlled by Player 2. We study three models of this game on oriented graphs which differ based on the kind of moves the players can make. These models are (i) the normal cop model, where both cops and robber can only move along the direction of the arcs; (ii) the strong cop model, where the cops can move along or against the direction of the arcs while the robber can only move along them; and (iii) the weak cop model, where the robber can move along or against the direction of the arcs while the cops can only move along them. The cop number is the minimum number of cops required to capture the robber in a graph, and a graph is cop-win if its cop number is 1.We begin our study by comparing the cop number in these three models for oriented graphs. For the normal cop model, we show that there exist strongly connected oriented graphs having high girth, high minimum degree, and high cop number. We also characterize the cop-win graphs in various graph classes like transitive-triangle-free, outerplanar, and subcubic graphs. For the strong cop model, we construct graphs with unbounded cop number, and also study the cop number of grids, outerplanar, and planar graphs. For the weak cop model, we characterize the cop-win graphs.

The Bridge Graphs Partition Dimension

Finding the partition dimension of graph is still an open problem in the graph theory. Therefore, several researchers investigate the problem in several operations of the graph. For example, the partition dimension of corona product, cartesian product, subdivision operation has been published by several researchers. Let G1, G2 be two connected graphs. We present the partition dimension of a graph G which is obtained from two graphs G1( G2 with a linking a vertex in G1 to one vertex in G2 (a bridge in graph G). This paper is devoted to find the upper bound of partition dimension of the connected two graphs by a bridge.

Capture-time extremal cop-win graphs

We investigate extremal graphs related to the game of Cops and Robbers. We focus on graphs where a single cop can catch the robber; such graphs are called cop-win. The capture time of a cop-win graph is the minimum number of moves the cop needs to capture the robber. We consider graphs that are extremal with respect to capture time, i.e. their capture time is as large as possible given their order. We give a new characterization of the set of extremal graphs. For our alternative approach we assign a rank to each vertex of a graph, and then study which configurations of ranks are possible. We partially determine which configurations are possible, enough to prove some further extremal results. We leave a full classification as an open question.