Graph Theory and applications to physics, chemistry, big data and vehicular traffic data

The Stone Age or lithic stage is the period of prehistory that goes from the stage where human beings began to develop stone tools until the discovery and usage of metals. Wood, bones and other materials were also used (horns, baskets, ropes, leather, etc.), but stone (and, in particular, various conchoidal fracture rocks, such as flint, quartz, quartzite, obsidian) was used to make tools and weapons, cutting or percussion.  (1)

Beyond the use of the stone as tools what was very important was the decision to join these stones with sticks by means of threads. Thus, with the ingenuity of the harvester hunter joining threads, the bases where stablished for what in the 21st century we call the Theory of Graphs whose precursor was precisely Leonard Euler.

In 1736, when he lived in Prussia, Euler solved the problem known as the Problem of the Seven Bridges of Königsberg. In the village of Königsberg, in Prussia, there is an island called Kneiphof, surrounded by two branches of the Pregel River: There are seven bridges, a, b, c, d, e, f, and g crossing the two branches, as shown in figure:

The question is whether a person can walk in such a way that he can cross each of these bridges once, but not more than once. Many insisted that this was impossible and others had doubts, and no one was able to say that it was possible. Euler continued his work by formulating a general theory that solves the particular problem and gives rise to a whole new branch of mathematics, called graph theory.  

The origin of the word graph is Greek and its etymological meaning is «trace» (from Greek graphs: drawing, image). The Theory of Graphs, a branch of Topology, is the study of mathematical structures that are used to model relationships between objects in a collection.

Graphs model connections in a network and are widely applicable to a variety of physical, biological, and information systems. You can use graphs to model the neurons in a brain, the flight patterns of an airline and in many more such as:

• Communication networks, design, routing.

• Vehicle distribution and routing problems.• Production planning.

• Traffic networks

• Demonstration of theorems

.• Verification and testing of programs

• VLSI Very Large Scale Integrated Systems

• Code Decryption

• Software Engineering• Databases

• Computational Biology

The structure of a graph consists of «nodes» and «edges». Each node represents an entity, and each edge represents a connection between two nodes.

In mathematics and computer science, a graph is a set of objects called vertices or nodes joined by links called edges or arcs, which allow to represent binary relationships between elements of a set. They are object of study of graph theory. 

Graphs in Chemistry and Physics

Each chemical formula defines a compound from the different atoms it contains. In 1811, the French chemist Joseph Louis Gay-Lussac (1778-1850) observed that there were different substances that had the same chemical composition, so that their atoms would have to be linked in different ways.

In 1857, the British polymath Arthur Cayley (1821-1895) and great expert in graph theory decided to use his mathematical knowledge in the study of isomers of alkanes of the formula CnH2n + 2.

Cayley wanted to list all possible configurations (2). His way of approaching the problem was to assign to each alkane a tree that is, a graph, a set of vertices and edges joining some of them, in which two vertices are connected by exactly one path of edges. Given an alkane, Cayley ignored hydrogen atoms, and constructed a tree by taking carbon atoms as vertices and as edges the bonds between them. His theory depended on the number of ‘centers’ in the alkane.

Graphs in architecture

The road system of a town or city can be represented by a network in which the usual network analysis of flows and capacities can be applied (3). However, in the early stages of designing such road systems, it is often important to consider the spatial layout of the roads and the spatial scheme they define. The sets of possibilities for road networks can be considered in the same way as for plant models, but now the edges of flat graphs represent roads. Different interpretations can be given to the different characteristics of the graph, and ornamentation operations can be modified to incorporate details of road junctions. Rectangular dissections, in particular, can be considered as road systems displayed on a grid, in which roads define the boundaries of rectangular blocks.

Social Networks, Big Data and the Google Page Rank algorithm

Computer networks can be connected in multiple ways and all of them give rise to a type of graph.

Graphs are a valuable tool in the social sciences, especially in sociology, anthropology, economy, communication, social psychology, etc. studies that analyze social networks: a social structure is represented by nodes of a graph (people, organizations, communities , Groups …) and the edges between nodes indicate the relevant relations (organization, economic independence, levels of decision, communications …).

In 1998, Brin and Page [19] introduced the notion of PageRank for the Google search algorithm on the Web (4). Unlike the usual methods in pattern comparison used earlier in data retrieval, the novelty of PageRank depends entirely on the underlying graphical web to determine the «importance» of a web page.

Although PageRank is originally designed For web graphics, the concept and definitions work well for any chart.

In fact, PageRank has become a valuable tool for examining the correlations of vertex pairs (or pairs of subsets) on any given graph and therefore leads to many applications in graph theory.

In graph theory there are many notions, such as distances (typically number of jumps needed to reach a vertex from another), cuts (ie subsets of vertices / edges that separate a part of the graph from the rest), flows That is, combinations of paths for routing between given vertices), and so on.

Traffic and vehicular chaos

Quantitative operations research techniques, with special emphasis on their application to the analysis and operation of transportation systems (urban, air, sea and road transportation, and cargo collection and delivery systems) and the planning and design of transportation systems. (5) Logistics oriented urban services (emergency care for firefighters and police, medical emergencies, urgent repair services). Study of functions of random variables, geometric probability, queuing and multiserver theory, spatial localization theory, network analysis and graph theory, and application of simulation methods. (6)

References:

1. The Stone Age, Wikipedia

2. Arthur Cayley, graph theory and chemical isomers.

3. Claudi Alsina. Metro Maps and Neural Networks. Graphic Schema Theory.

4. Graph Theory in the Information Age Fan Chung.

5. Polytechnic University of Valencia. Department of Applied Mathematics.

6. Emphasis on transit and transportation. Colombian School of Engineering Julio Garavito.

Fernando Jiménez Motte Ph.D (c) EE, MSEE, BSEE

CEO of NEUROMORPHIC TECHNOLOGIES NT Robotics, Control Systems, Artificial Intelligence AI

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