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.

Let \(g \in H(\mathcal{B})\), \(g(0) = 0\) and \(\varphi\) be a holomorphic self-map of the unit ball \(\mathbb{B}\) in \(\mathbb{C}^n\). The following integral-type operator

\[I_\varphi^g(f)(z) = \int_{0}^{1} {\mathcal{R}f(\varphi(tz))}{g(tz)}\frac{ dt}{t}, \quad f \in H(\mathbb{B}),z\in \mathbb{B},\]

was recently introduced by S. Stević and studied on some spaces of holomorphic functions on \(\mathbb{B}\), where \(\mathcal{R}f(z) = \sum_{k=1}^n z_k \frac{\partial f}{\partial z_k}(z)\) is the radial derivative of \(g\). The boundedness and compactness of this operator from generally weighted Bloch spaces to Bloch-type spaces on \(\mathbb{B}\) are investigated in this note.

We start by proving that the Henson graphs \(H_n\), \(n \geq 3\) (the homogeneous countable graphs universal for the class of all finite graphs omitting the clique of size \(n\)), are retract rigid. On the other hand, we provide a full characterization of retracts of the complement of \(H_3\). Further, we prove that each countable partial order embeds in the natural order of retractions of the complements of Henson graphs. Finally, we show that graphs omitting sufficiently large null subgraphs omit certain configurations in their endomorphism monoids.

Combining integration method with series rearrangement,we establish several closed formulae for Gauss hypergeometric series with four free parameters, which extend essentially the related results found recently by Elsner \((2005).\)

In this paper, we study the global behavior of the nonnegative equilibrium points of the difference equation

\(x_{n+1} = \frac{Ax_{n-2l}}{B+C \prod\limits_{i=0}^{2k}x_{n-i}}, n=0,1,\ldots ,\)

where \(A\), \(B\), \(C\) are nonnegative parameters, initial conditions are nonnegative real numbers, and \(k\), \(l\) are nonnegative integers, \(l \leq k\). Also, we derive solutions of some special cases of this equation.