In this paper, we present a new structure for chordal graph. We have also given the algorithm for MCS(Maximal Cardinality Search) and lexicographic BFS(Breadth First Search) which is used in two linear time and space algorithm. Also we discuss how to build a clique tree of a chordal graph and the other is simple recognition procedure of chordal graphs.Chordal graphs have been considered as the intersection graphs of subtrees of a tree. Chordal graphs are often represented by a clique tree. The structure of clique tree does not only appeared in the graph theory literature, but in context of ascyclic database schemes and in the context of spare matrix computations too.Chordal graphs can also be characterized using Perfect Elimination Orderings (PEO). A vertex is simplicial if and only if its neighbourhood is a complete subgraph. An elimination ordering x , x ,...xn 1 2 is perfect if and only if each xi is simplicial in the subgraph indued by i n x ,......x A new structure namely the clique graph is introduced .In this paper some graph properties of this structure are studied with regard to clique trees. And the clique graph is justified as being the optimal structure containing all clique trees of a chordal graph.