基于图谱的一些不变量的研究

图谱理论是代数图论和组合矩阵论共同关注的一个重要研究课题，且在理论化学、分子生物学和复杂网络等领域有广泛的应用。在图谱理论及其应用的研究中，一些基于图谱的不变量扮演了非常重要的角色。本项目主要研究其中三类图谱不变量：图的最大度与邻接谱半径之差、图的 Resolvent Estrada 指数以及图的拉普拉斯特征值幂和。图的最大度与邻接谱半径之差被用来度量图的非正则性，对它的研究在某种意义上是对图谱理论的经典结论的一种拓展，有比较强的理论意义。图的 Resolvent Estrada 指数与图的拉普拉斯特征值幂和的研究属于图谱理论与理论化学、复杂网络的交叉研究领域，可以为相关学科的进一步研究提供更有效的数学工具和方法，有比较好的应用价值。

The theory of graph spectra is an important research topic in both algebraic graph theory and combinatorial matrix theory, and has been widely used in theoretical chemistry, molecular biology, complex network, and so on. In the theory of graph spectra and its application, some graph-spectrum-based invariants play a very important role. Our project mainly study such three spectral invariants: the difference between maximum degree and adjacency spectral radius, Resolvent Estrada index, and the sum of powers of Laplace eigenvalues of a graph. The difference between maximum degree and adjacency spectral radius of a graph could be used to measure the non-regular property of the graph. In a certain sense, our study on this spectral invariant extends a classical result in the theory of graph spectra, and has important theoretical significance. Our study on Resolvent Estrada index and the sum of powers of Laplace eigenvalues of a graph, belonging to the interdisciplinary research field which contains the theory of graph spectra, theoretical chemistry, and complex network, would provide some more effective mathematical tools and methods for further investigation in related disciplines, and has good application value.