explane me about linear graphs?

Question

in maths

Answer 1

In graph theory, the line graph L(G) of an undirected graph G is a graph such that

each vertex of L(G) represents an edge of G; and
any two vertices of L(G) are adjacent if and only if their corresponding edges share a common endpoint in G.
That is, it is the intersection graph of the edges of G, representing each edge by the set of its two endpoints. Thus, properties of edges in graphs can be translated into properties about vertices in line graphs; for instance, the size of a maximum independent set in a line graph is the same as the size of a maximum matching in the original graph. The line graph is also sometimes called the edge graph, the adjoint graph, the interchange graph, or the derived graph of G.

One of the earliest and most important theorems about line graphs is due to Hassler Whitney (1932), who proved that with one exceptional case the structure of any connected graph can be recovered completely from its line graph.

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