We consider configuration graphs with N vertices. The degrees of the vertices are independent identically distributed limited random variables. They are equal to the number of vertex semiedges that are numbered in an arbitrary order. The graph is constructed by joining all of the semiedges pairwise equiprobably to form edges. We study the subset of such random graphs under the condition that the sum of vertex degrees is known and it is equal to n. An important characteristic of the topology of a graph is the local clustering coefficient. We obtained the limit distributions of the local clustering coefficient and the number of triangles of each vertex as N and n tend to infinity. We also considered the limit behaviour of their mathematical expectations.