Connectivity and spanning trees of graphs

This dissertation focuses on connectivity, edge connectivity and edge-disjoint spanning trees in graphs and hypergraphs from the following aspects.;1. Eigenvalue aspect. Let lambda2(G) and tau( G) denote the second largest eigenvalue and the maximum number of edge-disjoint spanning trees of a graph G, respectively. Motivated by a question of Seymour on the relationship between eigenvalues of a graph G and bounds of tau(G), Cioaba and Wong conjectured that for any integers d, k ≥ 2 and a d-regular graph G, if lambda 2(G)) \u3c d -- 2k-1d+1 , then tau(G) ≥ k. They proved the conjecture for k = 2, 3, and presented evidence for the cases when k ≥ 4. We propose a more general conjecture that for a graph G with minimum degree delta ≥ 2 k ≥ 4, if lambda2(G) \u3c delta -- 2k-1d+1 then tau(G) ≥ k. We prove the conjecture for k = 2, 3 and provide partial results for k ≥ 4. We also prove that for a graph G with minimum degree delta ≥ k ≥ 2, if lambda2( G) \u3c delta -- 2k-1d +1 , then the edge connectivity is at least k. As corollaries, we investigate the Laplacian and signless Laplacian eigenvalue conditions on tau(G) and edge connectivity.;2. Network reliability aspect. With graphs considered as natural models for many network design problems, edge connectivity kappa\u27(G) and maximum number of edge-disjoint spanning trees tau(G) of a graph G have been used as measures for reliability and strength in communication networks modeled as graph G. Let kappa\u27(G) = max{lcub}kappa\u27(H) : H is a subgraph of G{rcub}. We present: (i) For each integer k \u3e 0, a characterization for graphs G with the property that kappa\u27(G) ≤ k but for any additional edge e not in G, kappa\u27(G + e) ≥ k + 1. (ii) For any integer n \u3e 0, a characterization for graphs G with |V(G)| = n such that kappa\u27(G) = tau( G) with |E(G)| minimized.;3. Generalized connectivity. For an integer l ≥ 2, the l-connectivity kappal( G) of a graph G is defined to be the minimum number of vertices of G whose removal produces a disconnected graph with at least l components or a graph with fewer than l vertices. Let k ≥ 1, a graph G is called (k, l)-connected if kappa l(G) ≥ k. A graph G is called minimally (k, l)-connected if kappal(G) ≥ k but ∀e ∈ E( G), kappal(G -- e) ≤ k -- 1. A structural characterization for minimally (2, l)-connected graphs and some extremal results are obtained. These extend former results by Dirac and Plummer on minimally 2-connected graphs.;4. Degree sequence aspect. An integral sequence d = (d1, d2, ···, dn) is hypergraphic if there is a simple hypergraph H with degree sequence d, and such a hypergraph H is a realization of d. A sequence d is r-uniform hypergraphic if there is a simple r- uniform hypergraph with degree sequence d. It is proved that an r-uniform hypergraphic sequence d = (d1, d2, ···, dn) has a k-edge-connected realization if and only if both di ≥ k for i = 1, 2, ···, n and i=1ndi≥ rn-1r-1 , which generalizes the formal result of Edmonds for graphs and that of Boonyasombat for hypergraphs.;5. Partition connectivity augmentation and preservation. Let k be a positive integer. A hypergraph H is k-partition-connected if for every partition P of V(H), there are at least k(| P| -- 1) hyperedges intersecting at least two classes of P. We determine the minimum number of hyperedges in a hypergraph whose addition makes the resulting hypergraph k-partition-connected. We also characterize the hyperedges of a k-partition-connected hypergraph whose removal will preserve k-partition-connectedness

l-connectivity, l-edge-connectivity and spectral radius of graphs

Let G be a connected graph. The toughness of G is defined as t(G)=min{\frac{|S|}{c(G-S)}}, in which the minimum is taken over all proper subsets S\subset V(G) such that c(G-S)\geq 2 where c(G-S) denotes the number of components of G-S. Confirming a conjecture of Brouwer, Gu [SIAM J. Discrete Math. 35 (2021) 948--952] proved a tight lower bound on toughness of regular graphs in terms of the second largest absolute eigenvalue. Fan, Lin and Lu [European J. Combin. 110 (2023) 103701] then studied the toughness of simple graphs from the spectral radius perspective. While the toughness is an important concept in graph theory, it is also very interesting to study |S| for which c(G-S)\geq l for a given integer l\geq 2. This leads to the concept of the l-connectivity, which is defined to be the minimum number of vertices of G whose removal produces a disconnected graph with at least l components or a graph with fewer than l vertices. Gu [European J. Combin. 92 (2021) 103255] discovered a lower bound on the l-connectivity of regular graphs via the second largest absolute eigenvalue. As a counterpart, we discover the connection between the l-connectivity of simple graphs and the spectral radius. We also study similar problems for digraphs and an edge version