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Showing posts from September, 2022

Mathematics - An Art of Thinking

"Mathematics is not about numbers, equations, computations, or algorithms: it is about understanding." — William Paul Thurston, American mathematician In a previous blog , I spoke a lot about what mathematics isn’t. However, I don’t think I spoke quite enough on what mathematics truly is. Here's how I began that blog. "When we’re young we begin by learning the steps to add – we’re given the rules and we must learn to repeat them – no questions asked. Why does carry forward work the way it does? Well, no questions asked." However, I believe that the essence of truly understanding mathematics lies in the questions you ask – and the first one, the most fundamental one, is also probably one of the hardest questions you would and could find – What really is Mathematics? Something that differentiates explaining what mathematics is from explaining a lot of other concepts is that mathematics was not really ever invented . If a young student angry about excessive math h

Kőnig-Egerváry theorem

Graph theory has been my most favourite thing to learn in maths and and in this blog i hope to spread the knowledge about Kőnig's theorem. It is advised that the readers are aware about basic graph theory terminologies including bipartite graphs. Before going on to the theorem i would like to go on about matchings and vertex cover which we are going to use in the theorem  Matchings A matching $M$ of a graph $G$ is a subset of the edges of a graph in which no two edge share a common vertex (non adjacent edges). For example :- The Matching $M$ over here is edge $\{ 3 - 5 , 1-2 , \}$ or $\{ 1 - 2 , 4 - 3 \}$ etc .  Maximum Matching is a matching that contains the largest possible number of edges for instance in the above example the maximum matching is 2 edges as there cannot be a subset of non adjacent edges having greater than 2 edges (Readers are advised to try so they can convince themselves) A Perfect Matching  is a matching that matches all vertices of the graph or in other sen

An introduction to Graph theory

Graph Theory is a branch of mathematics and computer science that deals with modelling various relationships using vertices and edges. Graph theory as a field is crucial today, and is used everywhere we go - from searching the internet to artificial intelligence, and from DNA sequencing to Google Maps. The power of graph theory emerges from its simplicity. Despite being a concept that is fairly simple to understand, it is an active field of research with hundreds of open problems which mathematicians work on today.  Introduction The Königsberg Bridge Problem The city of Königsberg is located on the Pregel river in Prussia. As shown in the image (a) below, the river divided the city into 4 landmasses which were connected by seven bridges. The citizens often wondered if it was possible to start from home, travel through the city crossing every bridge exactly once and return home. Exercise: Find a round trip crossing each bridge exactly once, or try to prove that no such trip exists. The