Let $G = (V, E)$ be a simple graph with vertex set $V(G) = \{v_1, v_2, \ldots, v_n\}$ and edge set $E(G)$. Let $A(G)$ be the adjacency matrix of graph $G$ and also let $D(G)$ be the diagonal matrix with degrees of the vertices on the main diagonal. The Laplacian matrix of $G$ is $L(G)=D(G)-A(G)$. Among all eigenvalues of the Laplacian matrix $L(G)$ of a graph $G$, the most studied is the second smallest, called the algebraic connectivity $(a(G))$ of a graph $G$ [9]. Let $\alpha(G)$ be the independence number of graph $G$. Recently, it was conjectured that (see, [1]): $$a(G)+\alpha(G)$$ is minimum for $\overline{K_{p,\,q}\backslash \{e\}}$, where $e$ is any edge in $K_{p,\,q}$ and $\displaystyle{p=\Big\lfloor\frac{n}{2}\Big\rfloor}\,,\, q=\Big\lceil\frac{n}{2}\Big\rceil$ ($K_{p,\,q}$ is a complete bipartite graph). The aim of this paper is to show that this conjecture is true.

Recently a new graph convexity was introduced, arising from Steiner intervals in graphs that are a natural generalization of geodesic intervals. The Steiner tree of a set $W$ on $k$ vertices in a connected graph $G$ is a tree with the smallest number of edges in $G$ that contains all vertices of $W$. The Steiner interval $I(W)$ of $W$ consists of all vertices in $G$ that lie on some Steiner tree with respect to $W$. Moreover, a set $S$ of vertices in a graph $G$ is $k$-Steiner convex, denoted $g_k$-convex, if the Steiner interval $I(W)$ of every set $W$ on $k$ vertices is contained in $S$. In this paper we consider two types of local convexities. In particular, for every $k \gt 3$, we characterize graphs with $g_k$-convex closed neighborhoods around all vertices of the graph. Then we follow with a characterization of graphs with $g_4$-convex closed neighborhoods around all $g_4$-convex sets of the graph.

Let $F$, $G$, and $H$ be multigraphs. An $(F,G)$-decomposition of $H$ is an edge decomposition of $H$ into copies of $F$ and $G$ using at least one of each. For subgraphs $L$ and $R$ of $H$, an $(F,G)$-packing of $H$ with leave $L$ is an $(F,G)$-decomposition of $H-E(L)$, and an $(F,G)$-covering of $H$ with padding $R$ is an $(F,G)$-decomposition of $H+E(R)$. A maximum $(F,G)$-packing of $H$ is an $(F,G)$-packing of $H$ with a minimum leave. A minimum $(F,G)$-covering of $H$ is an $(F,G)$-covering of $H$ with a minimum padding. Let $k$ be a positive integer. A $k$-path, denoted by $P_k$, is a path on $k$ vertices. A $k$-star, denoted by $S_k$, is a star with $k$ edges. In this paper, we obtain a maximum $(P_{k+1},S_{k})$-packing of $\lambda K_n$, which has a leave of size $\lt k$, and a minimum $(P_{k+1},S_{k})$-covering of $\lambda K_n$, which has a padding of size $\lt k$. A similar result for $\lambda K_{n,n}$ is also obtained. As corollaries, necessary and sufficient conditions for the existence of $(P_{k+1},S_k)$-decompositions of both $\lambda K_n$ and $\lambda K_{n,n}$ are given.

Let $R(n)$ denote the number of representations of a large positive integer $n$ as the sum of two squares, two cubes and two sixth powers. In this paper, it is proved that the anticipated asymptotic formula of $R(n)$ fails for at most $O(\left( \log X \right)^{2+\varepsilon})$ positive integers not exceeding $X$. This is an improvement of T. D. Wooley's result which requires $O(\left( \log X \right)^{3+\varepsilon})$.

Basraoui and Sebbar showed that the Eisenstein series $E_2$ has infinitely many $\text{SL}_2(\Bbb{Z})$-inequivalent zeros in the upper half-plane $\Bbb{H}$, yet none in the standard fundamental domain $\mathfrak{F}$. They also found infinitely many such regions containing a zero of $E_2$ and infinitely many regions which do not have any zeros of $E_2$. In this paper we study the zeros of the quasi-modular form $E_2(z)+NE_2(Nz)$ of weight $2$ for $\Gamma_0^+(N)$.

Let $n$ be an integer. Then, it is well known that there are infinitely many imaginary quadratic fields with an ideal class group having a subgroup isomorphic to $\mathbb{Z}/n\mathbb{Z} \times \mathbb{Z}/n\mathbb{Z}$. Less is known for real quadratic fields, other than the cases that $n=3,5,$ or $7$, due to Craig [3] and Mestre [4, 5]. In this article, we will prove that there exist infinitely many real quadratic number fields with the ideal class group having a subgroup isomorphic to $\mathbb{Z}/n\mathbb{Z} \times \mathbb{Z}/n\mathbb{Z}$ In addition, we will prove that there exist infinitely many imaginary quadratic number fields with the ideal class group having a subgroup isomorphic to $\mathbb{Z}/n\mathbb{Z} \times \mathbb{Z}/n\mathbb{Z} \times \mathbb{Z}/n\mathbb{Z}$.

Consider the difference Riccati equation $f(z+1)=\frac{a(z)f(z)+b(z)}{c(z)f(z)+d(z)}$, where $a,~b,~c,~d$ are polynomials, we precisely estimate growth of meromorphic solutions. To the difference Riccati equation $f(z+1)=\frac{A(z)+f(z)}{1-f(z)}$, where $A(z)=\frac{m(z)}{n(z)},$ $m(z),~n(z)$ are irreducible nonconstant polynomials, we precisely estimate exponents of convergence of zeros and poles of meromorphic solutions $f(z)$, their differences $\Delta f(z)=f(z+1)-f(z)$ and divided differences $\frac{\Delta f(z)}{f(z)}$.

Under the assumptions that $p(t),\;q(t)$ are regularly varying functions satisfying condition $$\int_a^\infty\frac{dt}{p(t)^{\frac{1}{\alpha}}}=\infty,$$ existence and asymptotic form of regularly varying intermediate solutions are studied for a fourth-order quasilinear differential equation $$\left(p(t)|x''(t)|^{\alpha-1}\,x''(t)\right)^{\prime\prime}+q(t)|x(t)|^{\beta-1}\,x(t)=0,\quad \alpha \gt \beta\gt 0.$$ It is shown that the asymptotic behavior of all such solutions is governed by a unique explicit law.

Using periodic and anti-periodic eigenvalues, we present new criteria for guaranteeing the existence, uniqueness and asymptotic stability (in the sense of Lyapunov) of periodic solutions of a Duffing equation under conditions which are weaker than those used in the literature. The proof is based on the application of the existence theorem of Leray-Schauder type, Floquet theory, Lyapunov stability theory and some analytic techniques.

We show the existence of two nontrivial nonnegative solutions and infinitely many solutions for degenerate $p(x)$-Laplace equations involving concave-convex type nonlinearities with two parameters. By investigating the order of concave and convex terms and using a variational method, we determine the existence according to the range of each parameter. Some Caffarelli-Kohn-Nirenberg type problems with variable exponents are also discussed.

Using the Fenchel-Young duality and mountain pass geometry we derive a new multiple critical point theorem. In a finite dimensional setting it becomes three critical point theorem while in an infinite dimensional case we obtain the existence of at least two critical points. The applications to anisotropic problems show that one can obtain easily that all critical points are nontrivial.

In this paper, we study a class of multi-group SIRS epidemic models with nonlinear incidence rate which have cross patch infection between different groups. The basic reproduction number $\mathscr{R}_0$ is calculated. By using the method of Lyapunov functions, LaSalle's invariance principle, the theory of the nonnegative matrices and the theory of the persistence of dynamical systems, it is proved that if $\mathscr{R}_0\leq 1$ then the disease-free equilibrium is globally asymptotically stable, and if $\mathscr{R}_0\gt 1$ then the disease in the model is uniform persistent. Furthermore, when $\mathscr{R}_0\gt 1$, by constructing new Lyapunov functions we establish the sufficient conditions of the global asymptotic stability for the endemic equilibrium.

We obtain a unique continuation result for fractional Schrödinger operators with potential in Morrey spaces. This is based on Carleman inequalities for fractional Laplacians.

We derive closed form expressions for the Laplace transform of certain quadratic Brownian functionals based on the Ornstein-Uhlenbeck process, using both Fredholm determinants and PDE arguments. Classical and new bond pricing formulas in quadratic Brownian models are obtained as particular cases.

In this paper, we study the strong convergence of the Halpern type algorithms for a strongly quasi-nonexpansive sequence of operators. These results extend the results of Saejung [11]. Some applications in infinite family of firmly quasi-nonexpansive mappings, multiparameter proximal point algorithm, constraint minimization and subgradient projection are presented.