The center of a tree is a vertex with minimal eccentricity. The eccentricity of a vertex X in a tree G is the maximum distance between the vertex X and any other vertex of the tree. The maximum eccentricity is the tree diameter. If a tree has only one center, it is called Central Tree and if a tree has only more than one centers, it is called Bi-central Tree. Every tree is either central or bi-central.Algorithm to find centers and bi-centers of a treeStep 1 − Remove all the vertices of degree 1 from the given tree and also ... Read More

Bipartite Graph - If the vertex-set of a graph G can be split into two disjoint sets, V1 and V2 , in such a way that each edge in the graph joins a vertex in V1 to a vertex in V2 , and there are no edges in G that connect two vertices in V1 or two vertices in V2 , then the graph G is called a bipartite graph.Complete Bipartite Graph - A complete bipartite graph is a bipartite graph in which each vertex in the first set is joined to every single vertex in the second set. The ... Read More

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 More

To reach a conclusion on quantified statements, there are four rules of inference which are collectively called as Inference Theory of the Predicate Calculus.Table of Rules of InferenceRule of InferenceName$$\begin{matrix} \forall x P(x) \ \hline \therefore P(y) \end{matrix}$$Rule US: Universal Specification$$\begin{matrix} P(c) \text { for any c} \ \hline \therefore \forall x P(x) \end{matrix}$$Rule UG: Universal Generalization$$\begin{matrix} \exists x P(x) \ \hline \therefore P(c) \text { for any c} \ \end{matrix}$$Rule ES: Existential Specification$$\begin{matrix} P(c) \text { for any c} \ \therefore \exists x P(x) \end{matrix}$$Rule EG: Existential GeneralizationRule US: Universal Specification - From $(x)P(x)$, one can conclude $P(y)$.Rule US: ... Read More

Independent sets are represented in sets, in whichthere should not be any edges adjacent to each other. There should not be any common vertex between any two edges.there should not be any vertices adjacent to each other. There should not be any common edge between any two vertices.Independent Line SetLet 'G' = (V, E) be a graph. A subset L of E is called an independent line set of 'G' if no two edges in L are adjacent. Such a set is called an independent line set.ExampleLet us consider the following subsets −L1 = {a, b} L2 = {a, b} ... Read More

Independent sets are represented in sets, in whichthere should not be any edges adjacent to each other. There should not be any common vertex between any two edges.there should not be any vertices adjacent to each other. There should not be any common edge between any two vertices.Independent Vertex SetLet 'G' = (V, E) be a graph. A subset of 'V' is called an independent set of 'G' if no two vertices in 'S' are adjacent.ExampleConsider the following subsets from the above graphs −S1 = {e} S2 = {e, f} S3 = {a, g, c} S4 = {e, d}Clearly, S1 ... Read More

Adjacency MatrixAdjacency Matrix is used to represent a graph. We can represent directed as well as undirected graphs using adjacency matrices. Following are the key properties of an Adjacency matrix.PropertiesAn Adjacency Matrix A[V][V] is a 2D array of size V × V where V is the number of vertices in a undirected graph.If there is an edge between Vx to Vy then the value of A[Vx][Vy] = 1 and A[Vy][Vx]=1, otherwise the value will be zero.For a directed graph, if there is an edge between Vx to Vy, then the value of A[Vx][Vy]=1, otherwise the value will be zero.Adjacency Matrix ... Read More

To reach on a conclusion on quantified statements, there are four rules of inference which are collectively called as Inference Theory of the Predicate Calculus.Table of Rules of InferenceRule of InferenceName$$\begin{matrix} \forall x P(x) \ \hline \therefore P(y) \end{matrix}$$Rule US: Universal Specification$$\begin{matrix} P(c) \text { for any c} \ \hline \therefore \forall x P(x) \end{matrix}$$Rule UG: Universal Generalization$$\begin{matrix} \exists x P(x) \ \hline \therefore P(c) \text { for any c} \ \end{matrix}$$Rule ES: Existential Specification$$\begin{matrix} P(c) \text { for any c} \ \therefore \exists x P(x) \end{matrix}$$Rule EG: Existential GeneralizationRule US: Universal Specification - From $(x)P(x)$, one can conclude $P(y)$.Rule ... Read More

Throughput of a system refers to the rate of processing of a task thereby generating results. Ethernet is a set of technologies primarily used in LANs, whose primary data units are frames. The throughput of Ethernet is measured by the rate of successful delivery of frames over a communication channel.There are several methods for representing Ethernet throughput. The least ambiguous among them is calculation of channel efficiency. Channel efficiency, is the percentage of the net bit rate (in bits per second) of a channel that is actually communicated. Suppose that an Ethernet connection has a speed of 100 Mbps. But, ... Read More

Carrier Sense Multiple Access with Collision Detection (CSMA/CD) is a network protocol for carrier transmission that operates in the Medium Access Control (MAC) layer. It senses or listens whether the shared channel for transmission is busy or not, and defers transmissions until the channel is free.When more than one stations send their frames simultaneously, collision occurs. Back-off algorithm is a collision resolution mechanism which is commonly used to schedule retransmissions after collisions in Ethernet. The waiting time that a station waits before attempting retransmission of the frame is called as back off time.Algorithm of CSMA/CDStep 1) When a frame is ... Read More