Related papers: Spectral extremal problem for the odd prism

The Tur\'an number $\mathrm{ex}(n,H)$ of a graph $H$ is the maximum number of edges in an $n$-vertex graph which does not contain $H$ as a subgraph. The Tur\'{a}n number of regular polyhedrons was widely studied in a series of works due to…

For a set of graphs $\mathcal{F}$, let $\ex(n,\mathcal{F})$ and $\spex(n,\mathcal{F})$ denote the maximum number of edges and the maximum spectral radius of an $n$-vertex $\mathcal{F}$-free graph, respectively. Nikiforov ({\em LAA}, 2007)…

Let $C_{\ell}$ be the cycle of order ${\ell}$. The square of $C_{\ell}$, denoted by $C_{\ell}^2$, is obtained by joining all pairs of vertices with distance no more than two in $C_{\ell}$. A graph is called $F$-free if it does not contain…

Let $\mathcal{F}$ denote a set of graphs. A graph $G$ is said to be $\mathcal{F}$-free if it does not contain any element of $\mathcal{F}$ as a subgraph. The Tur\'an number is the maximum possible number of edges in an $\mathcal{F}$-free…

For a cycle $C_k$ on $k$ vertices, its $p$-th power, denoted $C_k^p$, is the graph obtained by adding edges between all pairs of vertices at distance at most $p$ in $C_k$. Let $\ex(n, F)$ and $\spex(n, F)$ denote the maximum possible number…

A graph is called $F$-free if it does not contain a copy of $F$. Let $G(r,s)$ denote a $K_{r+1}$-free graph of order $n$ with chromatic number at least $s$ that maximizes the spectral radius. Nikiforov [Linear Algebra Appl., 2007] proved…

Let ${\rm ex}(n,F)$ and ${\rm spex}(n,F)$ be the maximum size and maximum spectral radius of an $F$-free graph of order $n$, respectively. The value ${\rm spex}(n,F)$ is called the spectral extremal value of $F$. Nikiforov [J. Graph Theory…

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…

The classical spectral Tur\'{a}n problem is to determine the maximum spectral radius of an $\mathcal{F}$-free graph of order $n$. Zhai and Wang [Linear Algebra Appl, 437 (2012) 1641-1647] determined the maximum spectral radius of…

Let $C_{2k_1, 2k_2, \ldots, 2k_t}$ denote the graph obtained by intersecting $t$ distinct even cycles $C_{2k_1}, C_{2k_2}, \ldots, C_{2k_t}$ at a unique vertex. In this paper, we determine the unique graphs with maximum adjacency spectral…

Let ${\rm spex}(n,F)$ be the maximum spectral radius over all $F$-free graphs of order $n$, and ${\rm SPEX}(n,F)$ be the family of $F$-free graphs of order $n$ with spectral radius equal to ${\rm spex}(n,F)$. Given integers $n,k,p$ with…

The odd wheel $W_{2k+1}$ is the graph formed by joining a vertex to a cycle of length $2k$. In this paper, we investigate the largest value of the spectral radius of the adjacency matrix of an $n$-vertex graph that does not contain…

The classical spectral Tur\'{a}n problem is to determine the maximum spectral radius of an $\mathcal{F}$-free graph of order $n.$ Let $\mathcal{K}_{k}^{-}$ be the set of all unbalanced $K_k.$ In this paper, we focus on the spectral…

We describe the C_{2k+1}-free graphs on n vertices with maximum number of edges. The extremal graphs are unique except for n = 3k-1, 3k, 4k-2, or 4k-1. The value of ex(n,C_{2k+1}) can be read out from the works of Bondy, Woodall, and…

The $p$-spectral radius of a graph $G\ $of order $n$ is defined for any real number $p\geq1$ as \[ \lambda^{\left( p\right) }\left( G\right) =\max\left\{ 2\sum_{\{i,j\}\in E\left( G\right) \ }x_{i}x_{j}:x_{1},\ldots,x_{n}\in\mathbb{R}\text{…

Given a graph $H$, a graph is said to be $H$-free if it does not contain $H$ as a subgraph. A graph is color-critical when it has an edge whose removal leads to a reduction in its chromatic number. For a graph $H$ with a chromatic number of…

For a graph family $\mathcal F$, let $\mathrm{ex}(n,\mathcal F)$ and $\mathrm{spex}(n,\mathcal F)$ denote the maximum number of edges and maximum spectral radius of an $n$-vertex $\mathcal F$-free graph, respectively, and let…

Let $\emph{spex}_{\mathcal{OP}}(n,F)$ and $\emph{spex}_{\mathcal{P}}(n,F)$ be the maximum spectral radius over all $n$-vertex $F$-free outerplanar graphs and planar graphs, respectively. Define $tC_l$ as $t$ vertex-disjoint $l$-cycles,…

Given a planar graph family $\mathcal{F}$, let ${\rm ex}_{\mathcal{P}}(n,\mathcal{F})$ and ${\rm spex}_{\mathcal{P}}(n,\mathcal{F})$ be the maximum size and maximum spectral radius over all $n$-vertex $\mathcal{F}$-free planar graphs,…

Let $\mathscr{F}$ be a family of graphs. A graph $G$ is $\mathscr{F}$-free if $G$ does not contain any $F\in \mathcal{F}$ as a subgraph. The general Tur\'an number, denoted by $ex(n, H,\mathscr{F})$, is the maximum number of copies of $H$…