The Comparison On Eigenvalue Of The Connected Complementary Graph Of Graphs

As a branch of the graph theory,algebraic graph theory has some area to be studied to relate to use algebraic acknowledge and method to analysis some properties of adjacency matrix on graph and its complementary part.On the fundamental of that,we can study the characteristic polynomial and characteristic formula or characteristic value of the adjacency matrix on a kind of connected complementary graph.After Fiedler and Nikiforov give us spectral radius and its properties in 2010,spectral theory has been one of the important theory to deal with the question on the connected properties of complementary graph and comparing the characteristic value of them.Just on that fundamental,we can studying the relation between vertexes and edges on that graph,studying connected properties of the graph as well as studying the union of connected graph.On the fundamental of the theorem as well as the results recently,this essay give out the comparison of the minimum eigenvalue of the constitutional graph,its complementary parts are still to be connected.The main contents just listed as follows:The first part: introducing definition of the connected graph which has two pendant in one part,and analysis the character of it and complementary graph on construction.The second part: talking on three situation and the adjacency matrix of connected complementary graph which has two pendant in one part.Then calculate the least characteristic value blow three situations.The third part: giving out the comparison of the characteristic value of the connected and complementary graph,which has two pendant in some part.