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Computer Engineering Articles
Page 4 of 35
Finding the line covering number of a graph
The line covering number (also called the edge cover number) of a graph is the minimum number of edges required to cover all the vertices of the graph. An edge cover is a set of edges such that every vertex in the graph is an endpoint of at least one edge in the set. The line covering number is denoted by α1. Lower Bound Formula For a graph with n vertices, the line covering number has the following lower bound − α1 ≥ ⌈n / 2⌉ This is because each edge can cover ...
Read MoreFinding the matching number of a graph
A matching in a graph is a set of edges where no two edges share a common vertex. The matching number of a graph is the maximum number of edges in any matching − in other words, the largest set of edges you can select such that no vertex appears more than once. The matching number is denoted by β1. Upper Bound For a graph with n vertices, the matching number has the following upper bound − β1 ≤ ⌊n / 2⌋ This is because each edge in a matching uses exactly 2 vertices, and no vertex can ...
Read MoreFinding the number of spanning trees in a graph
The article is already well-structured and clean from the previous improvement. I'll replace the two JPG images with SVG diagrams and keep everything else intact. A spanning tree of a connected graph G is a subgraph that includes all the vertices of G and is a tree (connected with no cycles). A spanning tree with n vertices always has exactly n − 1 edges. A single graph can have multiple spanning trees, and finding the total count is a common problem in graph theory. How to Find Spanning Trees To find all spanning trees of a graph, ...
Read MoreEulerian Graphs
An Eulerian graph is a graph in which it is possible to traverse every edge exactly once and return to the starting vertex. This concept is named after the mathematician Leonhard Euler, who solved the famous Seven Bridges of Königsberg problem in 1736. Key Definitions Euler Graph − A connected graph G is called an Euler graph if there is a closed trail (circuit) that includes every edge of the graph G exactly once. Euler Path − An Euler path is a path that uses every edge of a graph exactly once. An Euler path starts and ...
Read MorePlanar Graphs
A planar graph is a graph that can be drawn in a plane without any of its edges crossing each other. Drawing a graph in the plane without edge crossings is called embedding the graph in the plane. Planar graphs are important in circuit design, map coloring, and network layout problems. Planar Graph A graph G is called planar if it can be drawn in a plane without any edges crossing. The same graph may have multiple drawings − some with crossings and some without. If at least one crossing-free drawing exists, the graph is planar. Example ...
Read MorePendent Vertex, Isolated Vertex and Adjacency of a graph
In graph theory, vertices are classified based on their degree (the number of edges connected to them). Two special types are pendent vertices (degree 1) and isolated vertices (degree 0). Adjacency describes the relationship between vertices or edges that share a common connection. Pendent Vertex A vertex with degree one is called a pendent vertex (also known as a leaf vertex). It has exactly one edge connected to it. Example edge ab a b deg(a) = 1 ...
Read MorePartitioning of a Set
A partition of a set S is a collection of n disjoint subsets P1, P2, ... Pn that satisfies the following three conditions − No subset is empty − Pi ≠ ∅ for all 0 < i ≤ n Union covers the entire set − P1 ∪ P2 ∪ ... ∪ Pn = S Subsets are mutually disjoint − Pa ∩ Pb = ∅ for a ≠ b S = { a, b, c, d, e, f, g, h } ...
Read MoreMininum spanning tree algorithms
A spanning tree of a weighted, connected, undirected graph G whose total edge weight is less than or equal to the weight of every other possible spanning tree is called a Minimum Spanning Tree (MST). The weight of a spanning tree is the sum of all the weights assigned to each of its edges. The two most popular algorithms to find an MST are Kruskal's Algorithm and Prim's Algorithm. Kruskal's Algorithm Kruskal's algorithm is a greedy algorithm that builds the MST by picking the smallest weighted edge at each step, as long as it does not form a ...
Read MoreMathematical Logic Statements and Notations
Mathematical logic uses formal notation to represent statements, determine their truth values, and reason about them systematically. The key building blocks are propositions, predicates, well-formed formulas, and quantifiers. Proposition A proposition is a declarative statement that has either a truth value "true" or a truth value "false". A proposition consists of propositional variables and connectives. We denote the propositional variables by capital letters (A, B, etc). The connectives connect the propositional variables. Some examples of propositions − "The sun rises in the east" − True "12 + 5 = 20" − False "x + 2 ...
Read MoreMathematical Foundation Introduction
Mathematics provides the theoretical foundation for computer science, engineering, and many other fields. It can be broadly classified into two categories − Continuous Mathematics − It is based upon the continuous number line or the real numbers. It is characterized by the fact that between any two numbers, there are almost always an infinite set of numbers. For example, a function in continuous mathematics can be plotted as a smooth curve without breaks. Discrete Mathematics − It involves distinct, separated values. Between any two points, there are a countable number of points. For example, if we have a ...
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