Starlike trees whose maximum degree exceed 4 are determined by their Q-spectra

Let $A(G)$ and $D(G)$ be the adjacency matrix and the degree matrix of a graph $G$, respectively. The matrix $D(G)+A(G)$ is called the signless Laplacian matrix of $G$. The spectrum of the matrix $D(G)+A(G)$ is called the Q-spectrum of $G$. A graph is said to be determined by its Q-spectrum if there is no other non-isomorphic graph with the same Q-spectrum. In this paper, we prove that all starlike trees whose maximum degree exceed $4$ are determined by their Q-spectra.     

Limit properties of monotone matrix functions

The basic objects in this paper are monotonically nondecreasing $n\times n$ matrix functions $D(·)$ defined on some open interval $?=(a\text{,}b)$ of $\mathbb{R}$ and their limit values $D(a)$ and $D(b)$ at the endpoints a and b which are, in general, selfadjoint relations in ${\mathbb{C}}^n$. Certain space decompositions induced by the matrix function $D(·)$ are made explicit by means of the limit values $D(a)$ and $D(b)$. They are a consequence of operator inequalities involving these limit values and the notion of strictness (or definiteness) of monotonically nondecreasing matrix functions. This treatment provides a geometric approach to the square-integrability of solutions of definite canonical systems of differential equations.     

The characteristic polynomial and the matchings polynomial of a weighted oriented graph

Let $G^{\sigma }$ be a weighted oriented graph with skew adjacency matrix $S(G^{\sigma })$. Then $G^{\sigma }$ is usually referred as the weighted oriented graph associated to $S(G^{\sigma })$. Denote by $?(G^{\sigma }\text{;}\lambda )$ the characteristic polynomial of the weighted oriented graph $G^{\sigma }$, which is defined as$?(G^{\sigma }\text{;}\lambda )=\mathit{det}(\lambda I_n-S(G^{\sigma }))=\sum _{i=0}^n{a}_i(G^{\sigma }){\lambda }^{n-i}\text{.}$In this paper, we begin by interpreting all the coefficients of the characteristic polynomial of an arbitrary real skew symmetric matrix in terms of its associated oriented weighted graph. Then we establish recurrences for the characteristic polynomial and deduce a formula on the matchings polynomial of an arbitrary weighted graph. In addition, some miscellaneous results concerning the number of perfect matchings and the determinant of the skew adjacency matrix of an unweighted oriented graph are given.     

Decomposition of integer-valued polynomial algebras

Let D be a commutative domain with field of fractions K, let A be a torsion-free D-algebra, and let B be the extension of A to a K-algebra. The set of integer-valued polynomials on A is $\mathrm{Int}(A)=\{f\in B[X]|f(A)?A\}$, and the intersection of $\mathrm{Int}(A)$ with $K[X]$ is ${\mathrm{Int}}_K(A)$, which is a commutative subring of $K[X]$. The set $\mathrm{Int}(A)$ may or may not be a ring, but it always has the structure of a left ${\mathrm{Int}}_K(A)$-module.A D-algebra A which is free as a D-module and of finite rank is called ${\mathrm{Int}}_K$-decomposable if a D-module basis for A is also an ${\mathrm{Int}}_K(A)$-module basis for $\mathrm{Int}(A)$; in other words, if $\mathrm{Int}(A)$ can be generated by ${\mathrm{Int}}_K(A)$ and A. A classification of such algebras has been given when D is a Dedekind domain with finite residue rings. In the present article, we modify the definition of ${\mathrm{Int}}_K$-decomposable so that it can be applied to D-algebras that are not necessarily free by defining A to be ${\mathrm{Int}}_K$-decomposable when $\mathrm{Int}(A)$ is isomorphic to ${\mathrm{Int}}_K(A){?}_{D}A$. We then provide multiple characterizations of such algebras in the case where D is a discrete valuation ring or a Dedekind domain with finite residue rings. In particular, if D is the ring of integers of a number field K, we show that an ${\mathrm{Int}}_K$-decomposable algebra A must be a maximal D-order in a separable K-algebra B, whose simple components have as center the same finite unramified Galois extension F of K and are unramified at each finite place of F. Finally, when both D and A are rings of integers in number fields, we prove that ${\mathrm{Int}}_K$-decomposable algebras correspond to unramified Galois extensions of K.     

Sharp bounds for the spectral radius of digraphs

Let $G=(V\text{,}E)$ be a digraph with n vertices and m arcs without loops and multiarcs. The spectral radius $\rho (G)$ of G is the largest eigenvalue of its adjacency matrix. In this paper, the following sharp bounds on $\rho (G)$ have been obtained.$\min \{\sqrt{t_i^{+}{t}_j^{+}}:(v_i\text{,}{v}_j)\in E\}?\rho (G)?\max \{\sqrt{t_i^{+}{t}_j^{+}}:(v_i\text{,}{v}_j)\in E\}$where G is strongly connected and $t_i^{+}$ is the average 2-outdegree of vertex $v_i$. Moreover, each equality holds if and only if G is average 2-outdegree regular or average 2-outdegree semiregular.     

Asymptotic expansions for Laplace transforms of Markov processes

Let ${\mu }^{?}$ be the probability measures on $D[0,T]$ of suitable Markov processes ${\{{\xi }_t^{?}\}}_{0\le t\le T}$ (possibly with small jumps) depending on a small parameter $?>0$, where $D[0,T]$ denotes the space of all functions on $[0,T]$ which are right continuous with left limits. In this paper we investigate asymptotic expansions for the Laplace transforms ${\int }_{D[0,T]}\exp ?\{{?}^{?1}F(x)\}{\mu }^{?}(dx)$ as $?\to 0$ for smooth functionals F on $D[0,T]$. This study not only recovers several well-known results, but more importantly provides new expansions for jump Markov processes. Besides several standard tools such as exponential change of measures and Taylor's expansions, the novelty of the proof is to implement the expectation asymptotic expansions on normal deviations which were recently derived in [13].     

On Ozaki's condition for p-valency

Let f be an analytic function in a convex domain $D?\mathbb{C}$. A well-known theorem of Ozaki states that if f is analytic in D, and is given by $f(z)=z^p+{\sum }_{n=p+1}^{\infty }{a}_n{z}^n$ for $z\in D$, and

$\mathfrak{Re}\{{\mathrm{e}}^{\mathrm{i}\alpha }{f}^{(p)}(z)\}>0,(z\in D),$

for some real α, then f is at most p-valent in D. Ozaki's condition is a generalization of the well-known Noshiro–Warschawski univalence condition. The purpose of this paper is to provide some related sufficient conditions for functions analytic in the unit disk $\mathbb{D}=\{z\in \mathbb{C}:|z|<1\}$ to be p-valent in $\mathbb{D}$, and to give an improvement to Ozaki's sufficient condition for p-valence when $z\in \mathbb{D}$.     

Topological invariants for semigroups of holomorphic self-maps of the unit disk

Let $({\phi }_t)$, $({?}_t)$ be two one-parameter semigroups of holomorphic self-maps of the unit disk $\mathbb{D}?\mathbb{C}$. Let $f:\mathbb{D}\to \mathbb{D}$ be a homeomorphism. We prove that, if $f°{?}_t={\phi }_t°f$ for all $t\ge 0$, then f extends to a homeomorphism of $\stackrel{￣}{\mathbb{D}}$ outside exceptional maximal contact arcs (in particular, for elliptic semigroups, f extends to a homeomorphism of $\stackrel{￣}{\mathbb{D}}$). Using this result, we study topological invariants for one-parameter semigroups of holomorphic self-maps of the unit disk.