Let \(k \geq 1\), \(l \geq 3\), and \(s \geq 5\) be integers. In \(1990\), Erdős and Faudree conjectured that if \(G\) is a graph of order \(4k\) with \(\delta(G) \geq 2k\), then \(G\) contains \(k\) vertex-disjoint \(4\)-cycles. In this paper, we consider an analogous question for \(5\)-cycles; that is to say, if \(G\) is a graph of order \(5k\) with \(\delta(G) \geq 3k\), then \(G\) contains \(k\) vertex-disjoint \(5\)-cycles? In support of this question, we prove that if \(G\) is a graph of order \(5k\) with \(\omega_2(G) \geq 6l – 2\), then, unless \(\overline{K_{l-2}} + K_{2l+1,2l+1} \subseteq G \subseteq K_{l-2} + K_{2l+1,2l+1}\), \(G\) contains \(l – 1\) vertex-disjoint \(5\)-cycles and a path of order \(5\), which is vertex-disjoint from the \(l – 1\) \(5\)-cycles. In fact, we prove a more general result that if \(G\) is a graph of order \(5k + 2s\) with \(\omega_2(G) \geq 6k + 2s\), then, unless \(\overline{K_{k}} + K_{2k+s,2k+s} \subseteq G \subseteq K_{k} + K_{2k+s,2k+s}\), \(G\) contains \(k+1\) vertex-disjoint \(5\)-cycles and a path of order \(2s – 5\), which is vertex-disjoint from the \(k + 1\) \(5\)-cycles. As an application of this theorem, we give a short proof for determining the exact value of \(\text{ex}(n,(k + 1)C_5)\), and characterize the extremal graph.

In this paper, we present the complex factorizations of the Jacobsthal and Jacobsthal Lucas numbers by determinants of tridiagonal matrices.

In this paper, we find families of \((0, -1, 1)\)-tridiagonal matrices whose determinants and permanents equal the negatively subscripted Fibonacci and Lucas numbers. Also, we give complex factorizations of these numbers by the first and second kinds of Chebyshev polynomials.

We classify all finite near hexagons which satisfy the following properties for a certain \(t_2 \in \{1,2,4\}\):(i) every line is incident with precisely three points;(ii) for every point \(x\), there exists a point \(y\) at distance \(3\) from \(x\);(iii) every two points at distance \(2\) from each other have either \(1\) or \(t_2 + 1\) common neighbours;(iv) every quad is big. As a corollary, we obtain a classification of all finite near hexagons satisfying (i), (ii) and (iii) with \(t_2\) equal to \(4\).

In this paper, we obtain the largest Laplacian spectral radius for bipartite graphs with given matching number and use them to characterize the extremal general graphs.

For integers \(k, \theta \leq 3\) and \(\beta \geq 1\), an integer \(k\)-set \(S\) with the smallest element \(0\) is a \((k; \beta, \theta)\)-free set if it does not contain distinct elements \(a_{i,j}\) (\(1 \leq i \leq j \leq \theta\)) such that \(\sum_{j=1}^{\theta -1}a_{i ,j} = \beta a_{i_\theta}\). The largest integer of \(S\) is denoted by \(\max(S)\). The generalized antiaverage number \(\lambda(k; \beta, \theta)\) is equal to \(\min\{\max(S) : S \text{ is a } (k^0; \delta, 0)\text{-free set}\}\). We obtain:(1) If \(\beta \notin \{\theta-2, \theta-1, \theta\}\), then \(\lambda(m; \beta, \theta) \leq (\theta-1)(m-2) + 1\); (2) If \(\beta \geq {\theta-1}\), then \(\lambda(k; \beta, \theta) \leq \min\limits_{k=m+n}\{\lambda(m;\beta,\theta)+\beta \lambda (n;\beta,\theta)+1\}\), where \(k =m+n \) with \(n>m\geq 3\) and \(\lambda(2n;\beta,\theta)\leq \lambda(n;\beta,\theta)(\beta+1)+\varepsilon\), for \(\varepsilon=1\) for \(\theta=3\) and \(\varepsilon=0\) otherwise.

A connected graph is highly irregular if the neighbors of each vertex have distinct degrees. We will show that every highly irregular tree has at most one nontrivial automorphism. The question that motivated this work concerns the proportion of highly irregular trees that are asymmetric, i.e., have no nontrivial automorphisms. A \(d\)-tree is a tree in which every vertex has degree at most \(d\). A technique for enumerating unlabeled highly irregular \(d\)-trees by automorphism group will be described for \(d \geq 4\) and results will be given for \(d = 4\). It will be shown that, for fixed \(d\), \(d \geq 4\), almost all highly irregular \(d\)-trees are asymmetric.

Combining with specific degrees or edges of a graph, this paper provides some new classes of upper embeddable graphs and extends the results in [Y. Huang, Y. Liu, Some classes of upper embeddable graphs, Acta Mathematica Scientia, \(1997, 17\)(Supp.): \(154-161\)].

A graph is called integral if all eigenvalues of its adjacency matrix are integers. In this paper, we investigate integral trees \(S(r;m_i) = S(a_1+a_2+\cdots+a_s;m_1,m_2,\ldots,m_s)\) of diameter \(4\) with \(s = 2,3\). We give a better sufficient and necessary condition for the tree \(S(a_1+a_2;m_1,m_2)\) of diameter \(4\) to be integral, from which we construct infinitely many new classes of such integral trees by solving some certain Diophantine equations. These results are different from those in the existing literature. We also construct new integral trees \(S(a_1+a_2+a_3;m_1,m_2,m_3) = S(a_1+1+1;m_1,m_2,m_3)\) of diameter \(4\) with non-square numbers \(m_2\) and \(m_3\). These results generalize some well-known results of P.Z. Yuan, D.L. Zhang \(et\) \(al\).

Zagreb indices are the best known topological indices which reflect certain structural features of organic molecules. In this paper we point out that the modified Zagreb indices are worth studying and present some results about product graphs.