What is a graph | What is a liner graph | linear graph in graph theory

A linear graph (simply a graph) G(V,E) consists of a set of objects V={v1,v2……} called vertices, and another set E={e1,e2……..} whose elements are called edges, such that each edge  is identified with an unordered pair(v_i,v_j) of vertices. The vertices v_i, v_j associated with each edge e_k are called end vertices of e_k.

Definition also permits an edge to be associated with a vertex   pair (v_i,v_i) such an edge having the same vertex as both its end vertices is called self loop. V₂ is having self loop.

The definition also permits more than one edge associated with a given pair of vertices, such edges are called parallel edges. Such as e₄ and e₅ of the following graph.

A graph that has neither self loop nor parallel edges is called a simple graph.

A graph is also called a linear complex, a 1-complex, or a one-dimensional complex. A vertex is also called as a node, a junction, a point, 0-cell, or a 0-simplex. Other term used for an edge is a branch, a line, an element, a 1-cell, and a 1-simplex. But we generally use graph, vertex, and edges.