Statement For

My research involves finding bounds of graph theoretic invariants, in particular the independence number of a graph; constructing algorithms for calculating values of these invariants; investigating the structure of independent sets of vertices in graphs; and identifying and programming successful heuristics for automating conjecture-making. of these conjectures have involved the independence number α of a graph, the maximum number of vertices which are pairwise nonadja-cent. For instance, Graffiti conjectured, and Chung proved, that the radius of a graph is no more than its independence number (r ≤ α) [4]. Another Graffiti conjecture led to Chung's proof that the average distance between two vertices in a graph is no more than its independence number (¯ d ≤ α) [3]. Following a conjecture of Graffiti, and with we have proved [14] and characterized the case of equality for the following lower and upper bounds bounds for the independence number of a graph in terms of its number C of cut vertices: C 2 + 1 ≤ α ≤ n − C 2 − 1 2 [14]. The independence number of a graph is an NP-hard invariant [12], and thus finding bounds on this invariant can lead to improvements in estimating the value of this number in cases where it cannot be directly computed, and also in improving algorithms for the computation of this number. Graffiti has also made conjectures about various forms of carbon, whose mathematical structure can be represented by a graph: fullerenes,