this T(G) is a weird graph since there are no edges between one original vertex and one original edge; are you sure there isn't also an edge between original edges and their adjacent vertices?

in any case, in an n-cycle, the numbers of edges and vertices are the same, so this T(G) is a graph consisting of two disjoint copies of the original graph (item (i) shows that the vertex part of T(G) is identical to G for any graph, and i think item (ii) shows that the edge part of T(G) is related to the dual hyper graph of G for any graph, and an n-cycle is self-dual)

Thank you sir

What is the applications of the IDEALS in real life situations

T(G) seems to be the union of G and its line graph.

Read some of the recent articles by Jack Edmonds. He delves into these kinds of situations.

Your definition of the total graph is incomplete. The vertex set of T(G) is the union of V(G) and E(G). The edges are as follows:

(i) If u and v are vertices which are adjacent in G, then uv is an edge of T(G),

(ii) If e and f are edges of G which share an endpoint in G, then ef is an edge of T(G),

(iii) If u is a vertex of G and e is an edge of G such that u is an endpoint of e, then ue is an edge of T(G). 

Alternatively, T(G) is the "square" of the graph obtained by subdividing every edge of G. For more information about total graphs and squares of graphs, see the links below.

http://mathworld.wolfram.com/TotalGraph.html

http://en.wikipedia.org/wiki/Graph_power

http://link.springer.com/chapter/10.1007%2FBFb0067353#page-1

The total graph of an n-cycle will be a 4-regular graph on 2n vertices.