本研究是一個3年期計劃，我們主要研究圖型的格林函數(Combinatorial Green’s Function) 所具有的各種性質，並將成果應用到電流網路（Electric Network)上來計算與討論電阻距離 (Resistance Distance)。本計畫將同時使用圖論分析技巧、矩陣分析技巧、電流網路分析技 巧、與機率方法來進行研究。並聚焦於如強正則圖（Strongly Regular Graph)、距離正則圖 (Distance Regular Graph)等結構性強的圖型與電流網路。在計畫的初期，我們將發展新觀點 與新技巧，來仔細檢視格林函數與電阻距離的已知重要古典定理與性質，給出不同、並更簡 潔的證明與推導。基於格林函數、拉普拉斯矩陣與電阻距離在學術上的重要與文獻的豐富， 我們需1.5年來執行這準備與發展新觀點與新技巧。接著，基於格林函數、拉普拉斯矩陣與 電阻距離在資訊科學、物理、社群網絡、化學、隨機漫步等領域上有大量的重要應用與大量 的文獻，我們需1.5年來將發展的觀點與技巧應用入不同領域，並進一步考慮較不具結構的圖類。 ;This is a three-year research project on discrete Green's functions and their applications to electrical network theory with emphasis on resistance distance on special classes of graphs including strongly regular graphs, distance regular graphs, Kneser graphs, and other less structured graphs. We seek to develop new shorter combinatorial (or algebraic) proofs of classical results in this area, extend previously known results, and study open questions and conjectures posed in the literature which we find of interest. The key challenge in this project is to uncover the relation between the discrete Green's functions, resistance distances and the properties of the underlying graph. We will use algebraic techniques, electrical network analysis, graph theoretic techniques and probabilistic arguments to reach the goals we set to achieve. We believe that our work in this project will provide deeper insights into spectral graph theory, random walks on graphs and electric network problems.
