几类极值谱图论问题的研究

谱极值图论是图谱理论和极值图论相互交叉后形成的研究领域，主要研究某些图类中谱参数的极值问题.在本项目中，合作双方将围绕着图的谱Turán型问题、饱和图及平面图的谱极值的刻画等方面中涉及的重要猜想和问题进行深入的研究，进一步挖掘图的结构性质.本项目主要研究以下三个问题：1.在图的谱Turán型问题方面，将重点刻画符合Cioabă-Desai-Tait猜想的谱极值图；2.在饱和图的谱极值方面，将重点研究Kim-Kim-Kostochka-O猜想，并考虑某些特殊图类(如路、圈及森林等)的饱和图的最小谱半径的值；3.平面图的谱极值方面，将着力解决Boots-Royle-Cao-Vince猜想.

Spectrum extremal graph theory is a research field formed by the intersection of graph spectrum theory and extremal graph theory. Its primary focus is on studying extremal problems of spectral parameters in certain classes of graphs. In this project, the collaborating parties will conduct in-depth research on important conjectures and problems related to spectrum Turán type problems, saturated graphs, and the characterization of spectral extremal properties of planar graphs. The main objectives of this project are as follows：1. Regarding the graph spectrum Turán type problems, the main emphasis will be on characterizing spectral extremal graphs that satisfy the Cioabă-Desai-Tait conjecture. 2. In terms of spectral extremal properties of saturated graphs, the main focus will be on studying the cases outlined in the Kim-Kim-Kostochka-O conjecture and evaluating the minimum spectral radius of saturated graphs belonging to certain special graph classes (such as paths, cycles, and forests). 3. The project will make significant efforts to address the Boots-Royle-Cao-Vince conjecture in terms of spectral extremal properties of planar graphs.