Tuyển tập các báo cáo nghiên cứu khoa học trên tạp chí toán học quốc tế đề tài: The characteristic polynomial of a graph is reconstructible from the characteristic polynomials of its vertex-deleted subgraphs and their complements. | The characteristic polynomial of a graph is reconstructible from the characteristic polynomials of its vertex-deleted subgraphs and their complements Elias M. Hagos emhagos@ Submitted June 10 1999 Accepted February 13 2000. Abstract The question of whether the characteristic polynomial of a simple graph is uniquely determined by the characteristic polynomials of its vertex-deleted subgraphs is one of the many unresolved problems in graph reconstruction. In this paper we prove that the characteristic polynomial of a graph is reconstructible from the characteristic polynomials of the vertex-deleted subgraphs of the graph and its complement. AMS Classification Numbers 05C60 05C50 1 Introduction Let G V E be a simple graph with a vertex set of at least three elements V 1 . ng. We denote by E G the set of its edges. A subgraph of G obtained by deleting vertex i and all its incident edges is called a vertex-deleted subgraph of G and is denoted by Gj. The collection of vertex-deleted subgraphs of G is known as its deck. The characteristic polynomial of G is the characteristic polynomial of its adjacency matrix A and is defined by PG x det xI A . We call the collection of the characteristic polynomials of the vertex-deleted subgraphs the polynomial deck of G and denote it by P G PG1 PG2 . PGng. In general a property of a graph is said to be reconstructible if it is uniquely determined by its deck. Tutte 11 proved that the characteristic polynomial of a graph is reconstructible from its deck. But is the full knowledge of the vertex-deleted subgraphs necessary to reconstruct the characteristic polynomial Gutman and Cvetkovic 6 first raised the still unresolved question 1 2 THE ELECTRONIC JOURNAL OF COMBINATORICS 7 2000 R12 of whether the polynomial deck of a simple graph on at least three vertices contains enough information to reconstruct its characteristic polynomial. Some results are reported in 2 8 10 . In this paper we prove that PG x is uniquely determined .
