A general theorem in spectral extremal graph theory

The extremal graphs $\mathrm{EX}(n,\mathcal F)$ and spectral extremal graphs $\mathrm{SPEX}(n,\mathcal F)$ are the sets of graphs on $n$ vertices with maximum number of edges and maximum spectral radius, respectively, with no subgraph in $\mathcal F$. We prove a general theorem which allows us to characterize the spectral extremal graphs for a wide range of forbidden families $\mathcal F$ and implies several new and existing results. In particular, whenever $\mathrm{EX}(n,\mathcal F)$ contains the complete bipartite graph $K_{k,n-k}$ (or certain similar graphs) then $\mathrm{SPEX}(n,\mathcal F)$ contains the same graph when $n$ is sufficiently large. We prove a similar theorem which relates $\mathrm{SPEX}(n,\mathcal F)$ and $\mathrm{SPEX}_\alpha(n,\mathcal F)$, the set of $\mathcal F$-free graphs which maximize the spectral radius of the matrix $A_\alpha=\alpha D+(1-\alpha)A$, where $A$ is the adjacency matrix and $D$ is the diagonal degree matrix

Problems in Extremal Combinatorics

This dissertation is divided into two major sections. Chapters 1 to 4 are concerned with Turán type problems for disconnected graphs and hypergraphs. In Chapter 5, we discuss an unrelated problem dealing with the equivalence of two notions of stationary processes. The Turán number of a graph H, ex(n,H), is the maximum number of edges in any n-vertex graph which is H-free. We discuss the history and results in this area, focusing particularly on the degenerate case for bipartite graphs. Let Pl denote a path on l vertices, and k*Pl denote k vertex-disjoint copies of Pl. We determine ex(n,k*P3) for n appropriately large, confirming a conjecture of Gorgol. Further, we determine ex(n,k*Pl) for arbitrary l, and n appropriately large. We provide background on the famous Erdös-Sós conjecture, and conditional on its truth we determine ex(n,H) when H is an equibipartite forest, for appropriately large n. In Chapter 4, we prove similar results in hypergraphs. We first discuss the related results for extremal numbers of hyperpaths, before proving the extremal numbers for multiple copies of a loose path of fixed length, and the corresponding result for linear paths. We extend this result to forests of loose hyperpaths, and linear hyperpaths. We note here that our results for loose paths, while tight, do not give the extremal numbers in their classical form; much more detail on this is given in Chapter 4. InChapter 5, we discuss two notions of stationary processes. Roughly, a process is a uniform martingale if it can be approximated arbitrarily well by a process in which the letter distribution depends only on a finite amount of the past. A random Markov process is a process with a coupled `look back\u27 time; that is, to determine the letter distribution, it suffices to choose a random look-back time, and then the distribution depends only on the past up to this time. Kalikow proved that on a binary alphabet, any uniform martingale is also a random Markov process. We extend this result to any finite alphabet

The signless Laplacian spectral radius of graphs without trees

Let $Q(G)=D(G)+A(G)$ be the signless Laplacian matrix of a simple graph of order $n$, where $D(G)$ and $A(G)$ are the degree diagonal matrix and the adjacency matrix of $G$, respectively. In this paper, we present a sharp upper bound for the signless spectral radius of $G$ without any tree and characterize all extremal graphs which attain the upper bound, which may be regarded as a spectral extremal version for the famous Erd\H{o}s-S\'{o}s conjecture.Comment: 12 page

Spectral extrema of {Kk+1,Ls}\{K_{k+1},\mathcal{L}_s\}-free graphs

For a set of graphs $\mathcal{F}$, a graph is said to be $\mathcal{F}$-free if it does not contain any graph in $\mathcal{F}$ as a subgraph. Let Ex$_{sp}(n,\mathcal{F})$ denote the graphs with the maximum spectral radius among all $\mathcal{F}$-free graphs of order $n$. A linear forest is a graph whose connected component is a path. Denote by $\mathcal{L}_s$ the family of all linear forests with $s$ edges. In this paper the graphs in Ex$_{sp}(n,\{K_{k+1},\mathcal{L}_s\})$ will be completely characterized when $n$ is appropriately large

Spectral extremal results on edge blow-up of graphs

The edge blow-up $F^{p+1}$ of a graph $F$ for an integer $p\geq 2$ is obtained by replacing each edge in $F$ with a $K_{p+1}$ containing the edge, where the new vertices of $K_{p+1}$ are all distinct. Let $ex(n,F)$ and $spex(n,F)$ be the maximum size and maximum spectral radius of an $F$-free graph of order $n$, respectively. In this paper, we determine the range of $spex(n,F^{p+1})$ when $F$ is bipartite and the exact value of $spex(n,F^{p+1})$ when $F$ is non-bipartite for sufficiently large $n$, which are the spectral versions of Tur\'{a}n's problems on $ex(n,F^{p+1})$ solved by Yuan [J. Combin. Theory Ser. B 152 (2022) 379--398]. This generalizes several previous results on $F^{p+1}$ for $F$ being a matching, or a star. Additionally, we also give some other interesting results on $F^{p+1}$ for $F$ being a path, a cycle, or a complete graph. To obtain the aforementioned spectral results, we utilize a combination of the spectral version of the Stability Lemma and structural analyses. These approaches and tools give a new exploration of spectral extremal problems on non-bipartite graphs

The α\alpha-index of graphs without intersecting triangles/quadrangles as a minor

The $A_{\alpha}$-matrix of a graph $G$ is the convex linear combination of the adjacency matrix $A(G)$ and the diagonal matrix of vertex degrees $D(G)$, i.e., $A_{\alpha}(G) = \alpha D(G) + (1 - \alpha)A(G)$, where $0\leq\alpha \leq1$. The $\alpha$-index of $G$ is the largest eigenvalue of $A_\alpha(G)$. Particularly, the matrix $A_0(G)$ (resp. $2A_{\frac{1}{2}}(G)$) is exactly the adjacency matrix (resp. signless Laplacian matrix) of $G$. He, Li and Feng [arXiv:2301.06008 (2023)] determined the extremal graphs with maximum adjacency spectral radius among all graphs of sufficiently large order without intersecting triangles and quadrangles as a minor, respectively. Motivated by the above results of He, Li and Feng, in this paper we characterize the extremal graphs with maximum $\alpha$-index among all graphs of sufficiently large order without intersecting triangles and quadrangles as a minor for any $0<\alpha<1$, respectively. As by-products, we determine the extremal graphs with maximum signless Laplacian radius among all graphs of sufficiently large order without intersecting triangles and quadrangles as a minor, respectively.Comment: 15 page

Spectral extremal problem on tt copies of ℓ\ell-cycle

Denote by $tC_\ell$ the disjoint union of $t$ cycles of length $\ell$. Let $ex(n,F)$ and $spex(n,F)$ be the maximum size and spectral radius over all $n$-vertex $F$-free graphs, respectively. In this paper, we shall pay attention to the study of both $ex(n,tC_\ell)$ and $spex(n,tC_\ell)$. On the one hand, we determine $ex(n,tC_{2\ell+1})$ and characterize the extremal graph for any integers $t,\ell$ and $n\ge f(t,\ell)$, where $f(t,\ell)=O(t\ell^2)$. This generalizes the result on $ex(n,tC_3)$ of Erd\H{o}s [Arch. Math. 13 (1962) 222--227] as well as the research on $ex(n,C_{2\ell+1})$ of F\"{u}redi and Gunderson [Combin. Probab. Comput. 24 (2015) 641--645]. On the other hand, we focus on the spectral Tur\'{a}n-type function $spex(n,tC_{\ell})$, and determine the extremal graph for any fixed $t,\ell$ and large enough $n$. Our results not only extend some classic spectral extremal results on triangles, quadrilaterals and general odd cycles due to Nikiforov, but also develop the famous spectral even cycle conjecture proposed by Nikiforov (2010) and confirmed by Cioab\u{a}, Desai and Tait (2022).Comment: 25 pages, one figur