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Computer Engineering Articles
Page 25 of 35
Cardinality of a Set
Cardinality of a set S, denoted by |S|, is the number of elements of the set. The number is also referred as the cardinal number. If a set has an infinite number of elements, its cardinality is ∞.Example − |{1, 4, 3, 5}| = 4, |{1, 2, 3, 4, 5, ....}| = ∞If there are two sets X and Y, |X| = |Y| denotes two sets X and Y having same cardinality. It occurs when the number of elements in X is exactly equal to the number of elements in Y. In this case, there exists a bijective function ‘f’ ...
Read MoreFunctions of Set
A Function assigns to each element of a set, exactly one element of a related set. Functions find their application in various fields like representation of the computational complexity of algorithms, counting objects, study of sequences and strings, to name a few. The third and final chapter of this part highlights the important aspects of functions.Function - DefinitionA function or mapping (Defined as f: X → Y) is a relationship from elements of one set X to elements of another set Y (X and Y are non-empty sets). X is called Domain and Y is called Codomain of function ‘f’.Function ...
Read MoreFinding the number of spanning trees in a graph
Problem StatementFind the number of spanning trees in the following graph.SolutionThe number of spanning trees obtained from the above graph is 3. They are as follows −These three are the spanning trees for the given graphs. Here the graphs I and II are isomorphic to each other. Clearly, the number of non-isomorphic spanning trees is two.
Read MoreFinding the number of regions in the graph
Problem StatementLet 'G' be a connected planar graph with 20 vertices and the degree of each vertex is 3. Find the number of regions in the graph.SolutionBy the sum of degrees theorem, 20 ∑ i=1 deg(Vi) = 2|E|20(3) = 2|E||E| = 30By Euler’s formula,|V| + |R| = |E| + 220+ |R| = 30 + 2|R| = 12Hence, the number of regions is 12.
Read MoreFinding the simple non-isomorphic graphs with n vertices in a graph
Problem StatementHow many simple non-isomorphic graphs are possible with 3 vertices?SolutionThere are 4 non-isomorphic graphs possible with 3 vertices. They are shown below.
Read MoreFinding the matching number of a graph
Problem StatementWhat is the matching number for the following graph?SolutionNumber of vertices = 9We can match only 8 vertices.Matching number is 4.
Read MoreFinding the line covering number of a graph
Problem StatementWhat is the line covering number for the following graph?SolutionNumber of vertices = |V| = n = 7Line covering number = (α1) ≥ ⌈ n / 2 ⌉ = 3α1 ≥ 3By using 3 edges, we can cover all the vertices.Hence, the line covering number is 3.
Read MoreFinding the chromatic number of complete graph
Problem StatementWhat is the chromatic number of complete graph Kn?SolutionIn a complete graph, each vertex is adjacent to is remaining (n–1) vertices. Hence, each vertex requires a new color. Hence the chromatic number Kn = n.
Read MoreEdges and Vertices of Graph
A graph is a set of points, called nodes or vertices, which are interconnected by a set of lines called edges. The study of graphs, or graph theory is an important part of a number of disciplines in the fields of mathematics, engineering and computer science.Graph TheoryDefinition − A graph (denoted as G = (V, E)) consists of a non-empty set of vertices or nodes V and a set of edges E. A vertex a represents an endpoint of an edge. An edge joins two vertices a, b and is represented by set of vertices it connects.Example − Let us ...
Read MoreDistance between Vertices and Eccentricity
Distance between Two VerticesIt is number of edges in a shortest path between Vertex U and Vertex V. If there are multiple paths connecting two vertices, then the shortest path is considered as the distance between the two vertices.Notation − d(U, V)There can be any number of paths present from one vertex to other. Among those, you need to choose only the shortest one.ExampleTake a look at the following graph −Here, the distance from vertex 'd' to vertex 'e' or simply 'de' is 1 as there is one edge between them. There are many paths from vertex 'd' to vertex ...
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