A note on the maximum number of triangles in a C5-free graph

We prove that the maximum number of triangles in a C5-free graph on n vertices is at most [Formula presented](1+o(1))n3/2, improving an estimate of Alon and Shikhelman [Alon, N. and C. Shikhelman, Many T copies in H-free graphs. Journal of Combinatorial Theory, Series B 121 (2016) 146-172]. © 2017 Elsevier B.V

A note on the maximum number of triangles in a C5-free graph

We prove that the maximum number of triangles in a C5-free graph on n vertices is at most 1/2√2(1+o(1))n3/2, improving an estimate of Alon and Shikhelman. © 2018 Wiley Periodicals, Inc

Maxima of the Q-index: forbidden 4-cycle and 5-cycle

This paper gives tight upper bounds on the largest eigenvalue q(G) of the signless Laplacian of graphs with no 4-cycle and no 5-cycle. If n is odd, let F_{n} be the friendship graph of order n; if n is even, let F_{n} be F_{n-1} with an edge hanged to its center. It is shown that if G is a graph of order n, with no 4-cycle, then q(G)<q(F_{n}), unless G=F_{n}. Let S_{n,k} be the join of a complete graph of order k and an independent set of order n-k. It is shown that if G is a graph of order n, with no 5-cycle, then q(G)<q(S_{n,2}), unless G=S_{n,k}. It is shown that these results are significant in spectral extremal graph problems. Two conjectures are formulated for the maximum q(G) of graphs with forbidden cycles.Comment: 12 page

Many TT copies in HH-free graphs

For two graphs $T$ and $H$ with no isolated vertices and for an integer $n$, let $ex(n,T,H)$ denote the maximum possible number of copies of $T$ in an $H$-free graph on $n$ vertices. The study of this function when $T=K_2$ is a single edge is the main subject of extremal graph theory. In the present paper we investigate the general function, focusing on the cases of triangles, complete graphs, complete bipartite graphs and trees. These cases reveal several interesting phenomena. Three representative results are: (i) $ex(n,K_3,C_5) \leq (1+o(1)) \frac{\sqrt 3}{2} n^{3/2},$ (ii) For any fixed $m$, $s \geq 2m-2$ and $t \geq (s-1)!+1$, $ex(n,K_m,K_{s,t})=\Theta(n^{m-\binom{m}{2}/s})$ and (iii) For any two trees $H$ and $T$, $ex(n,T,H) =\Theta (n^m)$ where $m=m(T,H)$ is an integer depending on $H$ and $T$ (its precise definition is given in Section 1). The first result improves (slightly) an estimate of Bollob\'as and Gy\H{o}ri. The proofs combine combinatorial and probabilistic arguments with simple spectral techniques

Generalized Tur\'an problems for disjoint copies of graphs

Given two graphs $H$ and $F$, the maximum possible number of copies of $H$ in an $F$-free graph on $n$ vertices is denoted by $ex(n,H,F)$. We investigate the function $ex(n,H,kF)$, where $kF$ denotes $k$ vertex disjoint copies of a fixed graph $F$. Our results include cases when $F$ is a complete graph, cycle or a complete bipartite graph.Comment: 18 pages. There was a wrong statement in the first version, it is corrected no