Advances in Pure and Applied Mathematics

APAM - ISSN 1869-6090 - © ISTE Ltd

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The Hosoya index of a graph $$$G$$$ is defined as the total number of matchings in $$$G$$$, including the empty edge set. In this paper, we study the maximum value of the Hosoya index. To this end, we introduce the notion of a "balanced forest of stars" and we characterize all forests of size $$$n$$$ and matching number $$$m$$$ that maximize the Hosoya index. We prove that this maximum is attained if and only if the forest is a balanced forest of stars.

This work involves the reconstruction in the sense of Ulam. Let $$$G$$$ and $$$G'$$$ be two digraphs with the same vertex set $$$V$$$. The digraphs $$$G$$$ and $$$G'$$$ are $$$\{-1\}$$$-hypomorphic if the induced subdigraphs $$$G-x$$$ and $$$G'-x$$$ are isomorphic for every vertex $$$x$$$. The digraph $$$G$$$ is $$$\{-1\}$$$-reconstructible if it is isomorphic to every digraph to which it is $$$\{-1\}$$$-hypomorphic. In a digraph $$$G$$$, a module is a vertex subset $$$M$$$ such that every vertex outside $$$M$$$ does not distinguish the vertices in $$$M$$$. For instance, $$$\emptyset$$$, $$$V(G)$$$, and the single-vertex sets are modules of $$$G$$$, called trivial modules. A module distinct from $$$V(G)$$$ is a proper module of $$$G$$$. A digraph is indecomposable if all its modules are trivial; otherwise, it is decomposable. Every indecomposable digraph with at least $$$3$$$ vertices is prime. A digraph is rigid if the unique automorphism of $$$G$$$ is the identity on $$$V(G)$$$.
In this paper, given an $$$n$$$-vertex digraph $$$G$$$, with $$$n\geq 7$$$, that can be expressed as a lexicographical sum over a prime and rigid digraph with at most $$$n-2$$$ vertices, we describe the digraphs that are $$$\{-1\}$$$-hypomorphic to $$$G$$$, and in particular we prove that $$$G$$$ is $$$\{-1\}$$$-reconstructible.

We characterize the boundedness and compactness of the composition operator $$$ C_{\psi}$$$, induced by an analytic self-map $$$\psi$$$ of the unit disk, acting from the weighted harmonic Bergman space $$$\mathcal{A}^{p}_{\beta}$$$, $$$1 ≤ p ≤ \infty$$$, into the harmonic Bloch spaces $$$\mathcal{B}^{\mu}_H$$$ and $$$\mathcal{B}^{\mu}_{H,0}$$$, where $$$\mu$$$ is a normal weight. The criteria are formulated both in terms of the action of $$$C_{\psi}$$$ on a family of normalized kernel-type test functions and in terms of explicit conditions on the symbol $$$\psi$$$. Analogous characterizations, with independent proofs, are obtained for $$$C_{\psi}$$$ acting from the harmonic Hardy space $$$\mathcal{H}^{q}$$$ into $$$\mathcal{B}^{\mu}_H$$$ and $$$\mathcal{B}^{\mu}_{H,0}$$$. The approach works directly with harmonic mappings and therefore also covers the endpoint cases $$$p=1$$$ and $$$q=1$$$, in which the analytic and co-analytic parts of a mapping in the space need not belong to the corresponding analytic space.

This work deals with traveling waves solutions of bistable reaction-diffusion systems in infinite cylinders. It is known that unique, stable traveling fronts exist for bistable systems. Here we are concerned with the traveling wave speed and how small perturbations affect the speed of the original problem. Using a mini-max variational formula, we prove an asymptotic expansion of the traveling wave speed and we determine the exact value of the first-order coefficient of this expansion.

This work aims to present a novel fashion of the multi-dimensional cubic functional equations and to investigate its Hyers-Ulam stability in the setting of Banach spaces on the restricted domains under a condition as (C). Moreover, a subset $$$\Omega\subset\mathbb R^d$$$ of Lebesgue measure zero satisfying the mentioned condition is constructed. Lastly, applying this construction and by means of the Baire category theorem, an asymptotic behavior for an approximate solution of the d-dimension cubic functional equations on real numbers, is presented.

We consider a compact Riemannian manifold $$$(M, g)$$$ of dimension $$$N \geq 3$$$ and $$$\Sigma$$$ a closed totally geodesic submanifold of dimension $$$1 \leq k \leq N-2$$$, and $$$h: M \to ℝ$$$ is a continuous function such that the linear operator $$$-Δ_g+h$$$ is coercive. We study existence of positive solutions $$$u \in H^1\left(M\right)$$$ to the following nonlinear PDE with two Hardy-Sobolev critical exponents:
(0.1) $$$ -\Delta_g u+h u=\lambda \rho_{\Sigma}^{-s_1} u^{2^*_{s_1}-1}+\rho_{\Sigma}^{-s_2} u^{2^*_{s_2}-1} \qquad \textrm{ in } (M, g)$$$
where $$$\lambda$$$ is a positive parameter, $$$0 < s_2 < s_1 < 2$$$, the $$$2^*_{s_i}:=\frac{2(N-s_i)}{N-2}$$$ $$$(i=1, 2)$$$ are two critical Hardy-Sobolev exponents and $$$\rho_\Sigma: \mathcal{M} \to ℝ$$$ is the distance function to $$$\Sigma$$$. In this paper, we give sufficient condition depending on the local geometries of the submanifold $$$\Sigma$$$ and the manifold $$$M$$$, for the existence of mountain pass solution to (0.1).

The main objective of this paper is to characterize a class of lattices, called splittable lattices, in the nilpotent semidirect product Lie group $$$G=\mathbb R^n\rtimes_\eta\mathbb R^m$$$. Here, $$$\eta$$$ denotes the representation of the abelian Lie group $$$\mathbb R^m$$$ in $$$GL_n(\mathbb R)$$$ defined by $$$\eta(t_{1},\dots,t_{m})=\exp{(\sum_{i=1}^mt_iN_i)}$$$ with $$$(N_i)_{1\leq i\leq m}$$$ is a set of pairwise commuting nilpotent matrices in $$$\mathbb Q^{n\times n}$$$. As an application we study the case of the nilpotent Lie group $$$F_{n,m}=\mathbb R^n\rtimes_\eta\mathbb R^m$$$ where $$$N_i=J^i$$$ with $$$J=\sum_{k=1}^{n-1}E_{k+1,k}$$$ where $$$E_{k+1,k}\in\mathbb R^{n\times n}$$$ denotes the matrix given by the Kronecker delta, namely, $$$\big(E_{k+1,k}\big)_{i,j}=\delta_{i,k+1}\delta_{j,k}$$$.

The study of manifolds equipped with special linear connections has become an important topic in contemporary Riemannian geometry. In this work, we investigate Kenmotsu manifolds endowed with a class of semi-symmetric non-metric connections that generalize several previously known connections in the literature. Particular attention is devoted to the geometric behavior of symmetric and skew-symmetric parallel tensor fields associated with such connections. We further examine the conditions under which these manifolds satisfy Ricci-symmetric, weakly symmetric, and weakly Ricci-symmetric properties. In addition, the existence and structure of Ricci solitons related to these non-metric connections are studied within the setting of Kenmotsu geometry.