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Let $spex(n,H_{minor})$ denote the maximum spectral radius of $n$-vertex $H$-minor free graphs. The problem on determining this extremal value can be dated back to the early 1990s. Up to now, it has been solved for $n$ sufficiently large and some special minors, such as $\{K_{2,3},K_4\}$, $\{K_{3,3},K_5\}$, $K_r$ and $K_{s,t}$. In this paper, we find some unified phenomena on general minors. Every graph $G$ on $n$ vertices with spectral radius $\rho\geq spex(n,H_{minor})$ contains either an $H$ minor or a spanning book $K_{\gamma_H}\nabla(n-\gamma_H)K_1$, where $\gamma_H=|H|-\alpha(H)-1$. Furthermore, assume that $G$ is $H$-minor free and $\Gamma^*_s(H)$ is the family of $s$-vertex irreducible induced subgraphs of $H$, then $G$ minus its $\gamma_H$ dominating vertices is $\Gamma^*_{\alpha(H)+1}(H)$-minor saturate, and it is further edge-maximal if $\Gamma^*_{\alpha(H)+1}(H)$ is a connected family. As applications, we obtain some known results on minors mentioned above. We also determine the extremal values for some other minors, such as flowers, wheels, generalized books and complete multi-partite graphs. Our results extend some conjectures on planar graphs, outer-planar graphs and $K_{s,t}$-minor free graphs. To obtain the results, we combine stability method, spectral techniques and structural analyses. Especially, we give an exploration of using absorbing method in spectral extremal problems.

Given a graph family $\mathcal{H}$ with $\min_{H\in \mathcal{H}}\chi(H)=r+1\geq 3$. Let ${\rm ex}(n,\mathcal{H})$ and ${\rm spex}(n,\mathcal{H})$ be the maximum number of edges and the maximum spectral radius of the adjacency matrix over all $\mathcal{H}$-free graphs of order $n$, respectively. Denote by ${\rm EX}(n,\mathcal{H})$ (resp. ${\rm SPEX}(n,\mathcal{H})$) the set of extremal graphs with respect to ${\rm ex}(n,\mathcal{H})$ (resp. ${\rm spex}(n,\mathcal{H})$). In this paper, we use a decomposition family defined by Simonovits to give a characterization of which graph families $\mathcal{H}$ satisfy ${\rm ex}(n,\mathcal{H})

The signless Laplacian spectral radius of a graph $G$, denoted by $q(G)$, is the largest eigenvalue of its signless Laplacian matrix. In this paper, we investigate extremal signless Laplacian spectral radius for graphs without short cycles or long cycles. Let $\mathcal{G}(m,g)$ be the family of graphs on $m$ edges with girth $g$ and $\mathcal{H}(m,c)$ be the family of graphs on $m$ edges with circumference $c$. More precisely, we obtain the unique extremal graph with maximal $q(G)$ in $\mathcal{G}(m,g)$ and $\mathcal{H}(m,c)$, respectively., Comment: 13 pages, one figure

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 figure

Let $G$ be a graph and $\lambda(G)$ be the spectral radius of $G$. A previous result due to Nikiforov [Linear Algebra Appl., 2009] in spectral graph theory asserted that every graph $G$ on $m\geq 10$ edges contains a 4-cycle if $\lambda(G)>\sqrt{m}$. Define $f(m)$ to be the minimum number of copies of 4-cycles in such a graph. A consequence of a recent theorem due to Zhai et al. [European J. Combin., 2021] shows that $f(m)=\Omega(m)$. In this article, by somewhat different techniques, we prove that $f(m)=\Theta(m^2)$. We left the solution to $\lim\limits_{m\rightarrow \infty} \frac{f(m)}{m^2}$ as a problem, and also mention other ones for further study., Comment: 14 pages

Motivated by the counting results for color-critical subgraphs by Mubayi [Adv. Math., 2010], we study the phenomenon behind Mubayi's theorem from a spectral perspective and start up this problem with the fundamental case of triangles. We prove tight bounds on the number of copies of triangles in a graph with a prescribed number of vertices and edges and spectral radius. Let $n$ and $m$ be the order and size of a graph. Our results extend those of Nosal, who proved there is one triangle if the spectral radius is more than $\sqrt{m}$, and of Rademacher, who proved there are at least $\lfloor\frac{n}{2}\rfloor$ triangles if the number of edges is more than that of 2-partite Tur\'an graph. These results, together with two spectral inequalities due to Bollob\'as and Nikiforov, can be seen as a solution to the case of triangles of a problem of finding spectral versions of Mubayi's theorem. In addition, we give a short proof of the following inequality due to Bollob\'as and Nikiforov [J. Combin. Theory Ser. B, 2007]: $t(G)\geq \frac{\lambda(G)(\lambda^2(G)-m)}{3}$ and characterize the extremal graphs. Some problems are proposed in the end., Comment: 15 pages; 1 figure; 1 table

A graph is outerplanar if it can be embedded in a plane such that all vertices lie on its outer face. The outerplanar Tur\'{a}n number of a given graph $H$, denoted by ${\rm ex}_{\mathcal{OP}}(n,H)$, is the maximum number of edges over all outerplanar graphs on $n$ vertices which do not contain a copy of $H$. In this paper, the outerplanar Tur\'{a}n numbers of cycles and paths are completely determined., Comment: 18 pages, 6 figures

In this paper, we give spectral conditions to guarantee the existence of two edge disjoint cycles and two cycles of the same length. These two results can be seen as spectral analogues of Erd\H{o}s and Posa's size condition and Erd\H{o}s' classic problem on non existence of two cycles of the same length. By using double leading eigenvectors skill, we further give spectral condition to guarantee the existence of $k$ edge disjoint triangles., Comment: 20 pages

Minors play an important role in extremal graph theory and spectral extremal graph theory. Tait [The Colin de Verdi\`{e}re parameter, excluded minors, and the spectral radius, J. Combin. Theory Ser. A 166 (2019) 42--58] determined the maximum spectral radius and characterized the unique extremal graph for $K_r$-minor free graphs of sufficiently large order $n$, he also made great progress on $K_{s,t}$-minor free graphs and posed a conjecture: Let $2\leq s\leq t$ and $n-s+1=pt+q$, where $n$ is sufficiently large and $1\leq q\leq t.$ Then $K_{s-1}\nabla (pK_t\cup K_q)$ is the unique extremal graph with the maximum spectral radius over all $n$-vertex $K_{s,t}$-minor free graphs. In this paper, Tait's conjecture is completely solved. We also determine the maximum spectral radius and its extremal graphs for $n$-vertex $K_{1,t}$-minor free graphs. To prove our results, some spectral and structural tools, such as, local edge maximality, local degree sequence majorization, double eigenvectors transformation, are used to deduce structural properties of extremal graphs., Comment: 27

The chromatic critical edge theorem of Simonovits states that for a given color critical graph $H$ with $\chi(H)=k+1$, there exists an $n_0(H)$ such that the Tur\'an graph $T_{n,k}$ is the only extremal graph with respect to $ex(n,H)$ provided $n \geq n_0(H)$. Nikiforov's pioneer work on spectral graph theory implies that the color critical edge theorem also holds if $ex(n,H)$ is replaced by the maximum spectral radius and $n_0(H)$ is an exponential function of $|H|$. We want to know which color critical graphs $H$ satisfy that $n_0(H)$ is a linear function of $|H|$. Previous graphs include complete graphs and odd cycles. In this paper, we find two new classes of graphs: books and theta graphs. Namely, we prove that every graph on $n$ vertices with $\rho(G)>\rho(T_{n,2})$ contains a book of size greater than $\frac{n}{6.5}$. This can be seen as a spectral version of a 1962 conjecture by Erd\H{o}s, which states that every graph on $n$ vertices with $e(G)>e(T_{n,2})$ contains a book of size greater than $\frac{n}{6}$. In addition, our result on theta graphs implies that if $G$ is a graph of order $n$ with $\rho(G)>\rho(T_{n,2})$, then $G$ contains a cycle of length $t$ for every $t\leq \frac{n}{7}$. This is related to an open question by Nikiforov which asks to determine the maximum $c$ such that every graph $G$ of large enough order $n$ with $\rho(G)>\rho(T_{n,2})$ contains a cycle of length $t$ for every $t\leq cn$., Comment: 17 pages

Nikiforov [Some inequalities for the largest eigenvalue of a graph, Combin. Probab. Comput. 179--189] showed that if $G$ is $K_{r+1}$-free then the spectral radius $\rho(G)\leq\sqrt{2m(1-1/r)}$, which implies that $G$ contains $C_3$ if $\rho(G)>\sqrt{m}$. In this paper, we follow this direction on determining which subgraphs will be contained in $G$ if $\rho(G)> f(m)$, where $f(m)\sim\sqrt{m}$ as $m\rightarrow \infty$. We first show that if $\rho(G)\geq \sqrt{m}$, then $G$ contains $K_{2,r+1}$ unless $G$ is a star; and $G$ contains either $C_3^+$ or $C_4^+$ unless $G$ is a complete bipartite graph, where $C_t^+$ denotes the graph obtained from $C_t$ and $C_3$ by identifying an edge. Secondly, we prove that if $\rho(G)\geq{\frac12+\sqrt{m-\frac34}}$, then $G$ contains pentagon and hexagon unless $G$ is a book; and if $\rho(G)>{\frac12(k-\frac12)+\sqrt{m+\frac14(k-\frac12)^2}}$, then $G$ contains $C_t$ for every $t\leq 2k+2$. In the end, some related conjectures are provided for further research., Comment: 21 pages, 6 pages. This paper has been accepted for publication in European Journal of Combinatorics

The planar Tur\'{a}n number of a given graph $H$, denoted by $ex_{\mathcal{P}}(n,H)$, is the maximum number of edges over all planar graphs on $n$ vertices that do not contain a copy of $H$ as a subgraph. Let $H_k$ be a friendship graph, which is obtained from $k$ triangles by sharing a common vertex. In this paper, we obtain sharp bounds of $ex_{\mathcal{P}}(n,H_k)$ and $ex_{\mathcal{P}}(n,K_1+P_{k+1})$ for $k\ge2$, which improves the results of Lan and Shi in Electron. J. Combin. 26 (2) (2019), \#P2.11.

Brualdi and Hoffman (1985) proposed the problem of determining the maximal spectral radius of graphs with given size. In this paper, we consider the Brualdi-Hoffman type problem of graphs with given matching number. The maximal $Q$-spectral radius of graphs with given size and matching number is obtained, and the corresponding extremal graphs are also determined.

Extremal problems concerning the number of complete subgraphs have a long story in extremal graph theory. Let $k_s(G)$ be the number of $s$-cliques in a graph $G$ and $m={{r_m}\choose s}+t_m$, where $0\le t_m\leq r_m$. Edr\H{o}s showed that $k_s(G)\le {{r_m}\choose s}+{{t_m}\choose{s-1}}$ over all graphs of size $m$ and order $n\geq r_m+1$. %Clearly, $K_{r_m}^{t_m}\cup (n-r_m-1)K_1$ is an extremal graph, where $K_{r_m}^{t_m}$ is the graph by joining a new vertex to $t_m$ vertices of $K_{r_m}$. It is natural to consider an improvement in connected situation: what is the maximum number of $s$-cliques over all connected graphs of size $m$ and order $n$? In this paper, the sharp upper bound of $k_s(G)$ is obtained and extremal graphs are completely characterized. The technique and the bound are different from those in general case. As an application, this result can be used to solve a question on spectral moment.

In this paper, we proceed on determining the minimum $q_{min}$ among the connected nonbipartite graphs on $n\geq 5$ vertices and with domination number $\frac{n+1}{3}<\gamma\leq \frac{n-1}{2}$. Further results obtained are as follows: $\mathrm{(i)}$ among all nonbipartite connected graph of order $n\geq 5$ and with domination number $\frac{n-1}{2}$, the minimum $q_{min}$ is completely determined; $\mathrm{(ii)}$ among all nonbipartite graphs of order $n\geq 5$, with odd-girth $g_{o}\leq5$ and domination number at least $\frac{n+1}{3}<\gamma\leq \frac{n-2}{2}$, the minimum $q_{min}$ is completely determined., Comment: arXiv admin note: substantial text overlap with arXiv:1707.07123

For a connected graph, the distance spectral radius is the largest eigenvalue of its distance matrix. In this paper, of all trees with both given order and fixed diameter, the trees with the minimal distance spectral radius are completely characterized., Comment: 10 pages,7 figures

Let $G$ be a graph with $n$ vertices and $\lambda_n(G)$ be the least eigenvalue of its adjacency matrix of $G$. In this paper, we give sharp bounds on the least eigenvalue of graphs without given pathes or cycles and determine the extremal graphs. This result gives spectral conditions for the existence of specified paths and cycles in graphs.