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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 Simonovits. In this paper, we shall present the exact Tur\'{a}n number of the prism $C_{2k+1}^{\square} $, which is defined as the Cartesian product of an odd cycle $C_{2k+1}$ and an edge $ K_2 $. Applying a deep theorem of Simonovits and a stability result of Yuan [European J. Combin. 104 (2022)], we shall determine the exact value of $\mathrm{ex}(n,C_{2k+1}^{\square})$ for every $k\ge 1$ and sufficiently large $n$, and we also characterize the extremal graphs. Moreover, in the case of $k=1$, motivated by a recent result of Xiao, Katona, Xiao and Zamora [Discrete Appl. Math. 307 (2022)], we will determine the exact value of $\mathrm{ex}(n,C_{3}^{\square} )$ for every $n$ instead of for sufficiently large $n$., Comment: 24 pages

Let $F_s$ be the friendship graph obtained from $s$ triangles by sharing a common vertex. For fixed $s\ge 2$ and sufficiently large $n$, the $F_s$-free graphs of order $n$ which attain the maximal spectral radius was firstly characterized by Cioab\u{a}, Feng, Tait and Zhang [Electron. J. Combin. 27 (4) (2020)],and later uniquely determined by Zhai, Liu and Xue [Electron. J. Combin. 29 (3) (2022)]. Recently, the spectral extremal problems was widely studied for graphs containing no $H$ as a minor. For instance, Tait [J. Combin. Theory Ser. A 166 (2019)], Zhai and Lin [J. Combin. Theory Ser. B 157 (2022)] solved the case $H=K_r$ and $H=K_{s,t}$, respectively. Motivated by these results, we consider the spectral extremal problems in the case $H=F_s$. We shall prove that $K_s \vee I_{n-s}$ is the unique graph that attain the maximal spectral radius over all $n$-vertex $F_s$-minor-free graphs. Moreover, let $Q_t$ be the graph obtained from $t$ copies of the cycle of length $4$ by sharing a common vertex. We also determine the unique $Q_t$-minor-free graph attaining the maximal spectral radius. Namely, $K_t \vee M_{n-t}$, where $M_{n-t}$ is a graph obtained from an independent set of order $n-t$ by embedding a matching consisting of $\lfloor \frac{n-t}{2}\rfloor$ edges.

Let $G$ be a graph of order $n$ and spectral radius be the largest eigenvalue of its adjacency matrix, denoted by $\mu(G)$. In this paper, we determine the unique graph with maximum spectral radius among all graphs of order $n$ without containing the $2$-power of a Hamilton cycle., Comment: 17 pages, 10 figures

A mixed graph is obtained by orienting some edges of a simple graph. The positive inertia index of a mixed graph is defined as the number of positive eigenvalues of its Hermitian adjacency matrix, including multiplicities. This matrix was introduced by Liu and Li, independently by Guo and Mohar, in the study of graph energy. Recently, Yuan et al. characterized the mixed graphs with exactly one positive eigenvalue. In this paper, we study the positive inertia indices of mixed graphs and characterize the mixed graphs with cut vertices having positive inertia index 2., Comment: 26 pages, 4 figures

Let $B_n$ be a linear polyomino chain with $n$ squares. Let $B_n^2$ be the graph obtained by the strong prism of a linear polyomino chain with $n$ squares, i.e. the strong product of $K_2$ and $B_n$. In this paper, explicit expressions for degree-Kirchhoff index and number of spanning trees of $B^2_n$ are determined, respectively. Furthermore, it is interesting to find that the degree-Kirchhoff index of $B^2_n$ is almost one eighth of its Gutman index., Comment: 16 pages, 2 figures. Comments are welcome!

The eccentricity matrix $\varepsilon(G)$ of a graph $G$ is constructed from the distance matrix of $G$ by keeping only the largest distances for each row and each column. This matrix can be interpreted as the opposite of the adjacency matrix obtained from the distance matrix by keeping only the distances equal to 1 for each row and each column. The $\varepsilon$-eigenvalues of a graph $G$ are those of its eccentricity matrix $\varepsilon(G)$. Wang et al \cite{e} proposed the problem of determining the maximum $\varepsilon$-spectral radius of trees with given order. In this paper, we consider the above problem of $n$-vertex trees with given diameter. The maximum $\varepsilon$-spectral radius of $n$-vertex trees with fixed odd diameter is obtained, and the corresponding extremal trees are also determined. The trees with least $\varepsilon$-eigenvalues in $[-2\sqrt{2},0)$ have been known. Finally, we determine the trees with least $\varepsilon$-eigenvalues in $[-2-\sqrt{13},-2\sqrt{2})$., Comment: 13pages, 4 figures

The first Zagreb index $M_{1}$ of a graph is defined as the sum of the square of every vertex degree, and the second Zagreb index $M_{2}$ of a graph is defined as the sum of the product of vertex degrees of each pair of adjacent vertices. In this paper, we study the Zagreb indices of bipartite graphs of order $n$ with $\kappa(G)=k$ (resp. $\kappa'(G)=s$) and sharp upper bounds are obtained for $M_1(G)$ and $M_2(G)$ for $G\in \mathcal{V}^k_n$ (resp. $\mathcal{E}^s_n$), where $\mathcal{V}^k_n$ is the set of bipartite graphs of order $n$ with $\kappa(G)=k$, and $\mathcal{E}^s_n$ is the set of bipartite graphs of order $n$ with $\kappa'(G)=s$., Comment: 9 pages, 4 figures