An asymptotically tight bound on the Q-index of graphs with forbidden cycles

Let G be a graph of order n and let q(G) be that largest eigenvalue of the signless Laplacian of G. In this note it is shown that if k>1 and q(G)>=n+2k-2, then G contains cycles of length l whenever 2<l<2k+3. This bound is asymptotically tight. It implies an asymptotic solution to a recent conjecture about the maximum q(G) of a graph G with no cycle of a specified length.Comment: 10 pages. Version 2 takes care of some mistakes in version

Maxima of the Q-index: graphs without long paths

This paper gives tight upper bound on the largest eigenvalue q(G) of the signless Laplacian of graphs with no paths of given order. The main ingredient of our proof is a stability result of its own interest, about graphs with large minimum degree and with no long paths. This result extends previous work of Ali and Staton.Comment: 10 page

Tur\`an numbers of Multiple Paths and Equibipartite Trees

The Tur\'an number of a graph H, ex(n;H), is the maximum number of edges in any graph on n vertices which does not contain H as a subgraph. Let P_l denote a path on l vertices, and kP_l denote k vertex-disjoint copies of P_l. We determine ex(n, kP_3) for n appropriately large, answering in the positive a conjecture of Gorgol. Further, we determine ex (n, kP_l) for arbitrary l, and n appropriately large relative to k and l. We provide some background on the famous Erd\H{o}s-S\'os conjecture, and conditional on its truth we determine ex(n;H) when H is an equibipartite forest, for appropriately large n.Comment: 17 pages, 13 figures; Updated to incorporate referee's suggestions; minor structural change

Improved bounds on the multicolor Ramsey numbers of paths and even cycles

We study the multicolor Ramsey numbers for paths and even cycles, $R_k(P_n)$ and $R_k(C_n)$, which are the smallest integers $N$ such that every coloring of the complete graph $K_N$ has a monochromatic copy of $P_n$ or $C_n$ respectively. For a long time, $R_k(P_n)$ has only been known to lie between $(k-1+o(1))n$ and $(k + o(1))n$. A recent breakthrough by S\'ark\"ozy and later improvement by Davies, Jenssen and Roberts give an upper bound of $(k - \frac{1}{4} + o(1))n$. We improve the upper bound to $(k - \frac{1}{2}+ o(1))n$. Our approach uses structural insights in connected graphs without a large matching. These insights may be of independent interest

The maximum number of PℓP_\ell copies in PkP_k-free graphs

Generalizing Tur\'an's classical extremal problem, Alon and Shikhelman investigated the problem of maximizing the number of $T$ copies in an $H$-free graph, for a pair of graphs $T$ and $H$. Whereas Alon and Shikhelman were primarily interested in determining the order of magnitude for large classes of graphs $H$, we focus on the case when $T$ and $H$ are paths, where we find asymptotic and in some cases exact results. We also consider other structures like stars and the set of cycles of length at least $k$, where we derive asymptotically sharp estimates. Our results generalize well-known extremal theorems of Erd\H{o}s and Gallai

Spectral radius of graphs forbidden C7C_7 or C6△C_6^{\triangle}

Let $C_k^{\triangle}$ be the graph obtained from a cycle $C_{k}$ by adding a new vertex connecting two adjacent vertices in $C_{k}$. In this note, we obtain the graph maximizing the spectral radius among all graphs with size $m$ and containing no subgraph isomorphic to $C_6^{\triangle}$. As a byproduct, we will show that if the spectral radius $\lambda(G)\ge1+\sqrt{m-2}$, then $G$ must contains all the cycles $C_i$ for $3\le i\le 7$ unless $G\cong K_3\nabla \left(\frac{m-3}{3}K_1\right)$.Comment: 11 pages, 1 figur

Extremal graphs without long paths and a given graph

For a family of graphs $\mathcal{F}$, the Tur\'{a}n number $ex(n,\mathcal{F})$ is the maximum number of edges in an $n$-vertex graph containing no member of $\mathcal{F}$ as a subgraph. The maximum number of edges in an $n$-vertex connected graph containing no member of $\mathcal{F}$ as a subgraph is denoted by $ex_{conn}(n,\mathcal{F})$. Let $P_k$ be the path on $k$ vertices and $H$ be a graph with chromatic number more than $2$. Katona and Xiao [Extremal graphs without long paths and large cliques, European J. Combin., 2023 103807] posed the following conjecture: Suppose that the chromatic number of $H$ is more than $2$. Then $ex\big(n,\{H,P_k\}\big)=n\max\big\{\big\lfloor \frac{k}{2}\big\rfloor-1,\frac{ex(k-1,H)}{k-1}\big\}+O_k(1)$. In this paper, we determine the exact value of $ex_{conn}\big(n,\{P_k,H\}\big)$ for sufficiently large $n$. Moreover, we obtain asymptotical result for $ex\big(n,\{P_k,H\}\big)$, which solves the conjecture proposed by Katona and Xiao.Comment: 16 pages, 6 conference