A Logical Connective is a symbol which is used to connect two or more propositional or predicate logics in such a manner that resultant logic depends only on the input logics and the meaning of the connective used.Generally there are five connectives which are −OR (∨)AND (∧)Negation/ NOT (¬)Implication / if-then (→)If and only if (⇔).OR (∨) − The OR operation of two propositions A and B (written as A ∨ B) is true if at least any of the propositional variable A or B is true.The truth table is as follows −ABA ∨ BTrueTrueTrueTrueFalseTrueFalseTrueTrueFalseFalseFalseAND (∧) − The AND operation ... Read More

A matching graph is a subgraph of a graph where there are no edges adjacent to each other. Simply, there should not be any common vertex between any two edges.MatchingLet 'G' = (V, E) be a graph. A subgraph is called a matching M(G), if each vertex of G is incident with at most one edge in M, i.e., deg(V) ≤ 1 ∀ V ∈ Gwhich means in the matching graph M(G), the vertices should have a degree of 1 or 0, where the edges should be incident from the graph G.Notation − M(G)ExampleIn a matching, if deg(V) = 1, ... Read More

A covering graph is a subgraph which contains either all the vertices or all the edges corresponding to some other graph. A subgraph which contains all the vertices is called a line/edge covering. A subgraph which contains all the edges is called a vertex covering.Line CoveringLet G = (V, E) be a graph. A subset C(E) is called a line covering of G if every vertex of G is incident with at least one edge in C, i.e., deg(V) ≥ 1 ∀ V ∈ Gbecause each vertex is connected with another vertex by an edge. Hence it has a minimum ... Read More

Kirchoff's theorem is useful in finding the number of spanning trees that can be formed from a connected graph.ExampleThe matrix 'A' be filled as, if there is an edge between two vertices, then it should be given as '1', else '0'.

IsomorphismIf two graphs G and H contain the same number of vertices connected in the same way, they are called isomorphic graphs (denoted by G ≅ H).It is easier to check non-isomorphism than isomorphism. If any of these following conditions occurs, then two graphs are non-isomorphic −The number of connected components are differentVertex-set cardinalities are differentEdge-set cardinalities are differentDegree sequences are differentExampleThe following graphs are isomorphic −HomomorphismA homomorphism from a graph G to a graph H is a mapping (May not be a bijective mapping) h: G → H such that − (x, y) ∈ E(G) → (h(x), h(y)) ∈ ... Read More

Tree is a discrete structure that represents hierarchical relationships between individual elements or nodes. A tree in which a parent has no more than two children is called a binary tree.Tree and its PropertiesDefinition − A Tree is a connected acyclic undirected graph. There is a unique path between every pair of vertices in G. A tree with N number of vertices contains (N-1) number of edges. The vertex which is of 0 degree is called root of the tree. The vertex which is of 1 degree is called leaf node of the tree and the degree of an internal ... Read More

Hamiltonian graph - A connected graph G is called Hamiltonian graph if there is a cycle which includes every vertex of G and the cycle is called Hamiltonian cycle. Hamiltonian walk in graph G is a walk that passes through each vertex exactly once.Dirac's Theorem - If G is a simple graph with n vertices, where n ≥ 3 If deg(v) ≥ {n}/{2} for each vertex v, then the graph G is Hamiltonian graph.Ore's Theorem - If G is a simple graph with n vertices, where n ≥ 2 if deg(x) + deg(y) ≥ n for each pair of non-adjacent ... Read More

There are various types of graphs depending upon the number of vertices, number of edges, interconnectivity, and their overall structure. We will discuss only a certain few important types of graphs in this chapter.Null GraphA graph having no edges is called a Null Graph.ExampleIn the above graph, there are three vertices named 'a', 'b', and 'c', but there are no edges among them. Hence it is a Null Graph.Trivial GraphA graph with only one vertex is called a Trivial Graph.ExampleIn the above shown graph, there is only one vertex 'a' with no other edges. Hence it is a Trivial graph.Non-Directed ... Read More

A graph can exist in different forms having the same number of vertices, edges, and also the same edge connectivity. Such graphs are called isomorphic graphs. Note that we label the graphs in this chapter mainly for the purpose of referring to them and recognizing them from one another.Isomorphic GraphsTwo graphs G1 and G2 are said to be isomorphic if −Their number of components (vertices and edges) are same.Their edge connectivity is retained.Note − In short, out of the two isomorphic graphs, one is a tweaked version of the other. An unlabelled graph also can be thought of as an ... Read More

Two graphs G1 and G2 are said to be homomorphic, if each of these graphs can be obtained from the same graph 'G' by dividing some edges of G with more vertices. Take a look at the following example −Divide the edge 'rs' into two edges by adding one vertex.The graphs shown below are homomorphic to the first graph.If G1 is isomorphic to G2, then G is homeomorphic to G2 but the converse need not be true.Any graph with 4 or less vertices is planar.Any graph with 8 or less edges is planar.A complete graph Kn is planar if and ... Read More