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.

pdf/eigenvalues-and-triangles-in-graphs-4tx749ht40.pdf

Eigenvalues and triangles in graphs

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

Spectral extrema of graphs with fixed size: Cycles and complete bipartite graphs

Spectral extrema of graphs: Forbidden hexagon

In this paper, we establish some sufficient conditions for a graph to be Hamilton-connected in terms of the edge number, the spectral radius and the signless Laplacian spectral radius of the graph. Furthermore, we also give some sufficient conditions for a graph to be traceable from every vertex in terms of the edge number, the spectral radius and the signless Laplacian spectral radius.

Some sufficient spectral conditions on Hamilton-connected and traceable graphs

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.

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

A complete solution to the Cvetković–Rowlinson conjecture

Spectral analogues of Moon–Moser's theorem on Hamilton paths in bipartite graphs

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