Full Binary Tree, Laplacian Eigenvalues ​​and Graph Merris Index

The theory of Laplacian eigenvalues of graphs is a basic research problem in field of graph theory, especially algebra graph theory, and gets more and more attention in recent years. Many articles on such topic have been released, including some results focusing on special graphs which have practical value in our life. By exact analysis of eigenvalues, we can describe characters of graphs. Also, when we study the eigenvalues of special graphs, some algebra properties are gained.In this paper, full binary tree in data structure is endued with new definition founding on graph theory, then eigenvalues of such graph are analysed systematically. Because of its tough symmetric structure, its degree sequence shows regular. So not only algebraic connectivity, but the whole eigenvalues have astonishing regular to be dug into. This paper goes deeply into such tree's structure using both depth search and breadth search, and gives recursive relationship separately. Through this, we discuss its eigenvalues problem, including regular on changing by depth added, multiplicities, integral eigenvalues and bounds estimation.Second, this paper discusses a special problem which was later defined as Merris index and Merris graphs resulting from Merris inequalities. The new definition is more integral and systematical. This paper gives some simple graphs' Merris indexes first. Further discussion on more general graphs are also made. Moreover, for the branches or union of Merris graphs, this paper gives their Merris properties.