Konigsberg Bridge Research Articles

Developing Phenomenological Sensitivity in Didactical Mathematics Through the Habit of Deep Observation in the Era of Industry 4.0

A teacher needs to have the sensitivity in seeing didactic phenomena. The sensitivity can function as a vehicle or context for mathematics learning. The Konigsberg Bridge, for example, can be described without looking at the picture since it became a part of the community story. Later on, the description of the Konigsberg Bridge can be used as a context in mathematics learning, especially those related to Graph Theory. This paper discusses the phenomena such as the Konigsberg Bridge, to make mathematics teachers understand the didactic phenomena easily. By having the sensitivity of the mathematically valuable phenomena for the mathematics teachers, they can contribute more positively to mathematics education. Keywords: phenomenological sensitivity, didactical matchematics, habit of deep observation

Indeks Padmakar-Ivan dan indeks Randic pada graf non-koprima dari grup bilangan bulat modulo

Graph theory, introduced by the Swiss mathematician Leonhard Euler in 1736, has played a pivotal role in solving real-world problems since its inception, notably exemplified by Euler's solution to the Konigsberg Bridge problem. Its applications extend to various domains, including scheduling, shortest path routing, and chemical structure representation. In chemistry, graphs are extensively used to depict molecular structures and chemical compounds, aiding in visualizing atomic connections and overall compound configurations. Topology indices, such as the Padmakar-Ivan (PI) and Randic indices, provide numerical values capturing chemical bonding relationships. Beyond chemical structures, these indices find applications in abstract algebraic graph representations. Recent research, exemplified by Husni et al.'s work on the harmonic and Gutman indices, explores these indices in coprime graphs of integer groups modulo prime power orders. Additionally, studies on non-coprime graphs of integer groups modulo reveal unique characteristics and invariants, shedding light on their structure. The non-coprime graph is a graph with two vertices said to be adjacent if the greatest common divisor (GCD) of their orders is not equal to one. This paper aims to investigate the topological indices, specifically the Padmakar-Ivan and Randic indices, in non-coprime graphs of integer groups modulo, adding depth to our understanding of their applicability and significance in abstract algebraic representations.
 Keywords: Graph theory, padmakar-ivan index, randic index, non-coprime graphsMSC2020: 05C09

Alternative solution for konigsberg bridge problem through the concept of matching

Alternative solution for konigsberg bridge problem through the concept of matching

A heuristic algorithm to find rupture degree in graphs

Since the problem of Konigsberg bridge was released in 1735, there have been many applications of graph theory in mathematics, physics, biology, computer science, and several fields of engineering. In particular, all communication networks can be modeled by graphs. The vulnerability is a concept that represents the reluctance of a network to disruptions in communication after a deterioration of some processors or communication links. Furthermore, the vulnerability values can be computed with many graph theoretical parameters. The rupture degree $r(G)$ of a graph $G=(V,E)$ is an important graph vulnerability parameter and defined as $r(G)=max\{\omega(G-S)- S -m(G-S):\omega(G-S)\geq2, S\subset V \}$, where $\omega(G-S)$ and $m(G-S)$ denote the number of connected components and the size of the largest connected component in the graph $G-S$, respectively. Recently, it has been proved that finding the rupture degree problem is $NP$-complete. In this paper, a heuristic algorithm to determine the rupture degree of a graph has been developed. Extensive computational experience on 88 randomly generated graphs ranging from 20\% to 90\% densities and from 100 to 200 vertices shows that the proposed algorithm is very effective.

Graph Routing Problem Using Euler’s Theorem and Its Applications

In this modern era, time and cases related to time is very important to us. For shortening time, Eulerian Circuit can open a new dimension. In computer science, social science and natural science, graph theory is a stimulating space for the study of proof techniques. Graphs are also effective in modeling a variety of optimization cases like routing protocols, network management, stochastic approaches, street mapping etc. Konigsberg Bridge Problem has seven bridges linked with four islands detached by a river in such a way that one can’t walk through each of the bridges exactly once and returning back to the starting point. Leonard Euler solved it in 1735 which is the foundation of modern graph theory. Euler’s solution for Konigsberg Bridge Problem is considered as the first theorem of Graph Theory which gives the idea of Eulerian circuit. It can be used in several cases for shortening any path. From the Konigsberg Bridge Problem to ongoing DNA fragmentation problem, it has its applications. Aiming to build such a dimension using Euler’s theorem and Konigsberg Bridge Problem, this paper presents about the history of remarkable Konigsberg Bridge Problem, Euler’s Explanation on it, an alternative explanation and some applications to Eulerian Circuit using graph routing and Fortran Coding of it.

Euler’s Königsberg: the explanatory power of mathematics

The present paper provides an analysis of Euler’s solutions to the Konigsberg bridges problem. Euler proposes three different solutions to the problem, addressing their strengths and weaknesses along the way. I put the analysis of Euler’s paper to work in the philosophical discussion on mathematical explanations. I propose that the key ingredient to a good explanation is the degree to which it provides relevant information. Providing relevant information is based on knowledge of the structure in question, graphs in the present case. I also propose computational complexity and logical strength as measures of relevant information.

Mathematics, Science, And Cultural Change

According to Roger Bacon (1214-1294), “Mathematics is the gate and key of the sciences. Neglect of Mathematics works injury to all knowledge, since he who is ignorant of it cannot know the other sciences or things of this world, and what is worse, men who are thus ignorant are unable to perceive their own ignorance and so do not seek a remedy”. Mathematics enables various branches of science draw implications of their observations and experimental findings. A computer is now an inevitable tool in science; its internal working rests on basic mathematics -arithematic to base2 Mathematics is at the heart of our important scientific theories; Newtonian mechanics, Electromagnetic theory of Maxwell, Einstein’s theory of relativity, quantum theory of Planck etc. Further, vast areas of mathematics grew out of efforts to find solutions to problems in science. We give two examples. The first one is about geometry, also sometimes referred to as the gift of the Nile. The story goes that in the 14th Century B.C., King Sesotris divided the land among Egyptians, each receiving a rectangle of same area, and was taxed accordingly. If anyone lost some of his land during the annual overflow of the Nile, he had to report the loss. An overseer would then be sent to measure the loss and make an abatement of tax. Some sources claim that this is the origin of geometry. The second example is more recent; it is about topology, whose origin is associated with a problem on a bridge. During the 18th century, in the German town of Konigsberg (now the Russian city of Kaliningrad), people enjoyed strolling along the banks of the Preger river, which meandered through the town, and was crossed by seven bridges which ran trom each bank of the river to two islands in the river, with a bridge joining the islands. One question asked is as follows: “How can you take a stroll so that you cross each of the seven bridges exactly once?”. Leonhard Eu!er ( 1707 -1783) converted this problem to one on vertices and networks in a diagram. He made general discoveries about networks, and found that the answer to the Konigsberg bridge problem, is in the negative, namely, that it is impossible to take a stroll so that each of the seven bridges is crossed only once! In the process of obtaining this result, he originated a new kind Of geometry, a geometry which does not depend on the size or shape of the figure, but about places, and how they are connected by arcs. Out of this grew the branch of mathematics called ‘Topology”.

A cost-time trade-off Königsberg bridge problem traversing all the seven bridges allowing repetition

An extended version of Konigsberg bridge problem is considered. After having split into two streams, Pregel River flows through the city of Konigsberg, now known as Kaliningrad, forming two islands. Seven bridges are built across the river providing links among the four land masses consisting of two islands, right and left banks of the river. Costs and times of traversing the bridges are given. The set of nondominated solutions of this problem is obtained providing the total cost and total time of nondominated paths starting from one land mass and returning to it after having traversed all the seven bridges allowing repetition. The problem is solved in two phases. In the first phase, the problem is represented by a graph whose nodes represent the land masses and edges represent the bridges. In the second phase, the set of nondominated solutions of the problem is obtained through generating and solving a sequence of prioritized bicriterion problems, all of whose nodes are of even degree. The generation of the sequence of prioritized bicriterion problems is done through augmenting the number of edges by duplicating some edges in the graph representing the problem in such a way that all the nodes in each of the prioritized bicriterion problems are of even degree; earlier all the nodes in the graph representing the problem were of odd degree. Now, since, each of the prioritized bicriterion problems has all the nodes of even degree, each of them is amenable to solution by Cycle finding and Fleury’s algorithms.

From L. Euler to D. König

Starting from the famous Konigsberg bridge problem which Euler described in 1736, we intend to show that some results obtained 180 years later by Konig are very close to Euler's discoveries.

About the cover: Euler and Königsberg's Bridges: A historical view

Graph theory almost certainly began when, in 1735, Leonhard Euler solved a popular puzzle about bridges. East Prussian city of Konigsberg (now Kalin- ingrad) occupies both banks of the River Pregel and an island, Kneiphof, which lies in the river at a point where it branches into two parts. There were seven bridges that spanned the various sections of the river, and the problem posed was this: could a person devise a path through Konigsberg so that one could cross each of the seven bridges only once and return home? Long thought to be impossible, the first mathematical demonstration of was presented by Euler to the members of the Petersburg Academy on August 26, 1735, and written up the following year under the title Solutio Problematis ad Geometriam Situs Pertinentis (The solution to a problem relating to the of position) (2) in the proceedings of the Petersburg Academy (the Commentarii). title page of volume appears on the cover of issue of the Bulletin. This story is well-known, and the illustrations in Euler's paper are often repro- duced in popular books on mathematics and in textbooks. Sandifer in (6) claims flatly that The Konigsberg Bridge Problem is Euler's most famous work, though scholars in other specialties (differential equations, complex analysis, calculus of variations, combinatorics, number theory, physics, naval architecture, music, . . . ) might disagree. N. L. Biggs, E. K. Lloyd, and R. J. Wilson in their history of graph theory (1) clearly view paper of Euler's as seminal and remark: The origins of graph theory are humble, even frivolous. Whereas many branches of mathematics were motivated by fundamental problems of calculation, motion, and measurement, the problems which led to the development of graph theory were often little more than puzzles, designed to test the ingenuity rather than to stimulate the imagina- tion. But despite the apparent triviality of such puzzles, they captured the interest of mathematicians, with the result that graph theory has become a subject rich in theoretical results of a surprising variety and depth. Euler provides only a neces- sary condition, not a sufficient condition, for solving the problem. But he does treat more than the original problem by beginning a generalization to two islands and four rivers, as is illustrated in the plate accompanying the original paper (Figure 1). In renowned paper Euler does not get around to stating the problem until the second page. On the first he states the reason for being interested in the problem— it was an example of a class of problems he attributes to Leibniz as belonging to something Leibniz called geometry of position. Euler says that this branch is concerned only with the determination of position and its properties; it does not

Introduction to the special section on graph algorithms in computer vision

N a letter to C. Huygens of 1679, G.W. Leibniz expressed his dissatisfaction with the standard coordinate treatment of geometric figures and maintained that we need yet another kind of analysis, geometric or linear, which deals directly with position, as algebra deals with magnitude (1). In fact, Leibniz initiated the study of the so-called geometry of positions (geometria situs) which, as L. Euler clearly put it in his famous 1736 Konigsberg bridges paper which had to mark the beginning of graph theory, concerned only with the determination of position, and its properties; it does not involve measurements nor calculations made with them (2). After about two centuries, this study developed into two of the richest branches of modern mathematics: graph theory and combinatorial topology. Mutatis mutandis, an analogous discontent is nowadays being felt among many researchers working in computer vision, a field that is currently dominated by purely geometric methods, who are increasingly making use of sophisticated graph-theoretic concepts, results, and algorithms. Indeed, graphs have long been an important tool in computer vision, especially because of their representational power and flexibility. However, there is now a renewed and growing interest toward explicitly formulating computer vision problems as graph problems. This is particularly advanta- geous because it allows vision problems to be cast in a pure, abstract setting with solid theoretical underpinnings and also permits access to the full arsenal of graph algorithms developed in computer science and operations research. Graph-theoretic problems which have proven to be relevant to computer vision include maximum flow, minimum spanning tree, maximum clique, shortest path, maximal common subtree/subgraph, etc. In addition, a number of fundamental techniques that were designed in the graph algorithms community have recently been applied to computer vision problems. Examples include spectral

The Konigsberg bridges problem generalized

The Konigsberg bridges problem generalized

The Seven Bridges of Königsberg and other Puzzles

The Seven Bridges of Königsberg and other Puzzles