Sharif Digital Repository

The reconstruction conjecture is an open and interesting problem that attracts some reserchers. Although some work toward this problem has been done; but no one have been able to solve it comlpetely. In this thesis, some new theorms are developed which extend the domain of reconstructible graphs. Also, a generalization of graph spectrum -called Alpha Spectrum- is introduced that is very effective in recognization of graphs  

In this thesis, we have shown that unique subgraphs of a graph have a key role in structure of the graph. Using unique subgraph which is called “anchor” here, the reconstruction of graphs is explained. Using anchor, we have shown that almost every n-vertex graph is determined by its 3log(n)-vertex subgraphs. In the second part of the thesis, a novel randomized algorithm is proposed for the graph isomorphism problem which is very simple and fast. It solves this problem with running time O(n^{2.373} \log(n)) for any pair of $n$-vertex graphs whose adjacency matrices are not strongly co-det. Strongly co-det pair of matrices have very special symmetric structure which can be disarranged to be...