Bo Ning

Bio: Bo Ning is an academic researcher from Nankai University. The author has contributed to research in topics: Induced subgraph & Conjecture. The author has an hindex of 11, co-authored 68 publications receiving 413 citations. Previous affiliations of Bo Ning include Tianjin University & Northwestern Polytechnical University.

Spectral analogues of Erdős’ and Moon–Moser’s theorems on Hamilton cycles

Abstract: In 1962, Erdős gave a sufficient condition for Hamilton cycles in terms of the vertex number, edge number and minimum degree of graphs which generalized Ore’s theorem. One year later, Moon and Moser gave an analogous result for Hamilton cycles in balanced bipartite graphs. In this paper, we present the spectral analogues of Erdős’ theorem and Moon–Moser’s theorem, respectively. Let be the class of non-Hamiltonian graphs of order n and minimum degree at least k. We determine the maximum (signless Laplacian) spectral radius of graphs in (for large enough n), and the minimum (signless Laplacian) spectral radius of the complements of graphs in . All extremal graphs with the maximum (signless Laplacian) spectral radius and with the minimum (signless Laplacian) spectral radius of the complements are determined, respectively. We also solve similar problems for balanced bipartite graphs and the quasi-complements.

Spectral radius and Hamiltonian properties of graphs

Abstract: Let be a graph with minimum degree . The spectral radius of , denoted by , is the largest eigenvalue of the adjacency matrix of . In this note, we mainly prove the following two results.(1) Let be a graph on vertices with . If , then contains a Hamilton path unless .(2) Let be a graph on vertices with . If , then contains a Hamilton cycle unless . As corollaries of our first result, two previous theorems due to Fiedler and Nikiforov and Lu et al. are obtained, respectively. Our second result refines another previous theorem of Fiedler and Nikiforov.

Eigenvalues and triangles in graphs

Abstract: Bollobas and Nikiforov (J. Combin. Theory Ser. B. 97 (2007) 859–865) conjectured the following. If G is a Kr+1-free graph on at least r+1 vertices and m edges, then by subdividing an edge. Both conditions are best possible. We conclude this paper with some open problems.

Rainbow triangles in edge-colored graphs

Abstract: Let G be an edge-colored graph. The color degree of a vertex v of G, is defined as the number of colors of the edges incident to v. The color number of G is defined as the number of colors of the edges in G. A rainbow triangle is one in which every pair of edges have distinct colors. In this paper we give some sufficient conditions for the existence of rainbow triangles in edge-colored graphs in terms of color degree, color number and edge number. As a corollary, a conjecture proposed by Li and Wang [H. Li and G. Wang, Color degree and heterochromatic cycles in edge-colored graphs, European J. Combin. 33 (2012) 1958-1964] is confirmed.

Extensions of the Erdős–Gallai theorem and Luo’s theorem

Abstract: The famous Erdős–Gallai theorem on the Turan number of paths states that every graph with n vertices and m edges contains a path with at least (2m)/n edges. In this note, we first establish a simple but novel extension of the Erdős–Gallai theorem by proving that every graph G contains a path with at least edges, where Nj(G) denotes the number of j-cliques in G for 1≤ j ≤ ω(G). We also construct a family of graphs which shows our extension improves the estimate given by the Erdős–Gallai theorem. Among applications, we show, for example, that the main results of [20], which are on the maximum possible number of s-cliques in an n-vertex graph without a path with l vertices (and without cycles of length at least c), can be easily deduced from this extension. Indeed, to prove these results, Luo [20] generalized a classical theorem of Kopylov and established a tight upper bound on the number of s-cliques in an n-vertex 2-connected graph with circumference less than c. We prove a similar result for an n-vertex 2-connected graph with circumference less than c and large minimum degree. We conclude this paper with an application of our results to a problem from spectral extremal graph theory on consecutive lengths of cycles in graphs.

Graphs and Matrices

Abstract: In order to store a graph or digraph in a computer, we need something other than the diagram or the formal definition. This something is the adjacency matrix, a matrix of O’s and l’s. The l’s correspond to the arcs of the digraph. Certain matrix operations will be seen to correspond to digraph concepts.

Problem (2)

Abstract: The worst case fall time is given by tf = 2.2 (2Rn•Cout + Rn•Cx) = 2.2(2*1 KΩ*8fF + 1KΩ*2fF) = 39.6ps b) The best-case rise time for a NAND gate occurs when both pMOS turn on and is given by tr = 2.2 (Rp/2) •Cout, where the total parallel resistance is Rp/2. = 2.2 * 2 KΩ/2 * 8 fF = 17.6 ps c) The worst-case rise time of a NOR gate is given by tr = 2.2 (2Rp•Cout + Rp•Cx) = 2.2(2*2 KΩ*8fF + 2KΩ*2fF) = 79.2 ps

Spectral analogues of Erdős’ and Moon–Moser’s theorems on Hamilton cycles

Abstract: In 1962, Erdős gave a sufficient condition for Hamilton cycles in terms of the vertex number, edge number and minimum degree of graphs which generalized Ore’s theorem. One year later, Moon and Moser gave an analogous result for Hamilton cycles in balanced bipartite graphs. In this paper, we present the spectral analogues of Erdős’ theorem and Moon–Moser’s theorem, respectively. Let be the class of non-Hamiltonian graphs of order n and minimum degree at least k. We determine the maximum (signless Laplacian) spectral radius of graphs in (for large enough n), and the minimum (signless Laplacian) spectral radius of the complements of graphs in . All extremal graphs with the maximum (signless Laplacian) spectral radius and with the minimum (signless Laplacian) spectral radius of the complements are determined, respectively. We also solve similar problems for balanced bipartite graphs and the quasi-complements.

Spectral radius and Hamiltonian properties of graphs

Abstract: Let be a graph with minimum degree . The spectral radius of , denoted by , is the largest eigenvalue of the adjacency matrix of . In this note, we mainly prove the following two results.(1) Let be a graph on vertices with . If , then contains a Hamilton path unless .(2) Let be a graph on vertices with . If , then contains a Hamilton cycle unless . As corollaries of our first result, two previous theorems due to Fiedler and Nikiforov and Lu et al. are obtained, respectively. Our second result refines another previous theorem of Fiedler and Nikiforov.

Eigenvalues and triangles in graphs

Abstract: Bollobas and Nikiforov (J. Combin. Theory Ser. B. 97 (2007) 859–865) conjectured the following. If G is a Kr+1-free graph on at least r+1 vertices and m edges, then by subdividing an edge. Both conditions are best possible. We conclude this paper with some open problems.