Edge-extremal Problems on the Relative Length of the Longest Paths and Cycles in Graphs

本论文主要研究图中关于最长路和最长圈的相对长度的边极值问题。论文共分三章。 在第一章，我们首先介绍了本文所涉及问题的基本定义、相关进展及主要结果。 在第二章和第三章，我们研究了关于图中最长路和最长圈的相对长度的边极值问题。我们将图G中最长路和最长圈的顶点个数分别记为p(G)和c(G)，定义两者的相对长度diff(G)=p(G)-c(G)。在第二章，我们证明了对于具有n个顶点的2-连通图G，其中，p(G)=p，如果p≥20并且e(G)>1/2(p-2)(n-7)+13，那么，我们有diff(G)≤1，由此推出，图G中的每一个最长圈都是一个控制圈。在p(G)=n时，我们给出图例说明所给的界是最...In this thesis, we mainly study edge-extremal problems on the relative length of the longest paths and cycles in graphs. This thesis consists of three chapters. In the first chapter, we introduce the background and main results of the research in the thesis. In the second and the third chapter, we present results on the relative length of the longest paths and cycles in graphs. For a graph G...学位：理学博士院系专业：数学科学学院数学与应用数学系_应用数学学号：1902006015316

Longest Paths in Circular Arc Graphs

As observed by Rautenbach and Sereni (arXiv:1302.5503) there is a gap in the proof of the theorem of Balister et al. (Longest paths in circular arc graphs, Combin. Probab. Comput., 13, No. 3, 311-317 (2004)), which states that the intersection of all longest paths in a connected circular arc graph is nonempty. In this paper we close this gap.Comment: 7 page

Sizing the length of complex networks

Among all characteristics exhibited by natural and man-made networks the small-world phenomenon is surely the most relevant and popular. But despite its significance, a reliable and comparable quantification of the question `how small is a small-world network and how does it compare to others' has remained a difficult challenge to answer. Here we establish a new synoptic representation that allows for a complete and accurate interpretation of the pathlength (and efficiency) of complex networks. We frame every network individually, based on how its length deviates from the shortest and the longest values it could possibly take. For that, we first had to uncover the upper and the lower limits for the pathlength and efficiency, which indeed depend on the specific number of nodes and links. These limits are given by families of singular configurations that we name as ultra-short and ultra-long networks. The representation here introduced frees network comparison from the need to rely on the choice of reference graph models (e.g., random graphs and ring lattices), a common practice that is prone to yield biased interpretations as we show. Application to empirical examples of three categories (neural, social and transportation) evidences that, while most real networks display a pathlength comparable to that of random graphs, when contrasted against the absolute boundaries, only the cortical connectomes prove to be ultra-short