Completely Characterizations Of Unicyclic And Bicyclic Graphs With Exactly Two Main Eigenvalues

Since Cvekoci`c([1]) proposes how to characterize graphs with exactly k(k≥2) main eigenvalues, there are some scholars to study this problem. Theunicyclic graphs with two main eigenvalues are characterized by Hou in [13]and the bicyclic graphs with two main eigenvalues are characterized by Shi in[15] and by Hu in [16] at the same time. Unfortunately there are some mistakesin the above three papers that lead to an incomplete characterization for thebicyclic graphs with two main eigenvalues. In this work, we fill up a leak inthe proof of Lemma 6 in [13] and supply an complete characterizations for allunicyclic and bicyclic connected graphs with exactly two main eigenvalues.In the first chapter, introduction, we give the basic definitions, symbolsand notations about eigenvalue and main eigenvalue of graph. We list someimportant known results about main eigenvalues.The second chapter consists of two sections. In the first section, we givea necessary and suficient conditions, given by Hagos, which is the basis ofcharacterizations of exactly two main eigenvalues in this paper. the secondsection, we prove some lemmas. At last, we give the completely argument ofthe unicyclic graph with exactly two main eigenvalues.The main purpose of the third chapter is to characterize the bicyclic graphwith exactly two main eigenvalues, Shi[15] and Hu[16] characterize the graphunder the condition of b≥0. And we will characterize the graph under thecondition of b < 0 and get two new graphs.