Hermitian quadratic matrix polynomials: Solvents and inverse problems

A monic quadratic Hermitian matrix polynomial $L(\lambda )$ can be factorized into a product of two linear matrix polynomials, say $L(\lambda )=(I\lambda -S)(I\lambda -A)$. For the inverse problem of finding a quadratic matrix polynomial with prescribed spectral data (eigenvalues and eigenvectors) it is natural to prescribe a right solvent A and then determine compatible left solvents S. This problem is explored in the present paper. The splitting of the spectrum between real eigenvalues and nonreal conjugate pairs plays an important role. Special attention is paid to the case of real-symmetric quadratic polynomials and the allocation of the canonical sign characteristics as well as the eigenvalues themselves.     

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.     

The sizes of optimal (n,4,λ,3) optical orthogonal codes

Let $\Phi (n,k,{\lambda }_a,{\lambda }_c)$ denote the largest possible size among all $(n,k,{\lambda }_a,{\lambda }_c)$-OOCs. An $(n,k,{\lambda }_a,{\lambda }_c)$-OOC with $\Phi (n,k,{\lambda }_a,{\lambda }_c)$ codewords is said to be optimal. In this paper, the exact value of $\Phi (n,4,\lambda ,3)$ is determined. Equivalently, the size of an optimal $(n,4,\lambda ,3)$ optical orthogonal code is calculated.     

On super 2-restricted and 3-restricted edge-connected vertex transitive graphs

Let $G=(V\left(G\right),E\left(G\right))$ be a simple connected graph and $F?E\left(G\right)$. An edge set $F$ is an $m$-restricted edge cut if $G?F$ is disconnected and each component of $G?F$ contains at least $m$ vertices. Let ${\lambda }^{\left(m\right)}\left(G\right)$ be the minimum size of all $m$-restricted edge cuts and ${\xi }_m\left(G\right)=min\{\left|\omega \left(U\right)\right|:\left|U\right|=m\text{ and }G\left[U\right]\text{ is connected}\}$, where $\omega \left(U\right)$ is the set of edges with exactly one end vertex in $U$ and $G\left[U\right]$ is the subgraph of $G$ induced by $U$. A graph $G$ is optimal-${\lambda }^{\left(m\right)}$ if ${\lambda }^{\left(m\right)}\left(G\right)={\xi }_m\left(G\right)$. An optimal-${\lambda }^{\left(m\right)}$ graph is called super $m$-restricted edge-connected if every minimum $m$-restricted edge cut is $\omega \left(U\right)$ for some vertex set $U$ with $\left|U\right|=m$ and $G\left[U\right]$ being connected. In this note, we give a characterization of super 2-restricted edge-connected vertex transitive graphs and obtain a sharp sufficient condition for an optimal-${\lambda }^{\left(3\right)}$ vertex transitive graph to be super 3-restricted edge-connected. In particular, a complete characterization for an optimal-${\lambda }^{\left(2\right)}$ minimal Cayley graph to be super 2-restricted edge-connected is obtained.     

On maximizing the fundamental frequency of the complement of an obstacle

Let $\mathrm{\Omega }?{\mathbb{R}}^n$ be a bounded domain satisfying a Hayman-type asymmetry condition, and let D be an arbitrary bounded domain referred to as an “obstacle”. We are interested in the behavior of the first Dirichlet eigenvalue ${\lambda }_1(\mathrm{\Omega }?(x+D))$.First, we prove an upper bound on ${\lambda }_1(\mathrm{\Omega }?(x+D))$ in terms of the distance of the set $x+D$ to the set of maximum points $x_0$ of the first Dirichlet ground state ${?}_{{\lambda }_1}>0$ of Ω. In short, a direct corollary is that if

(1)${\mu }_{\mathrm{\Omega }}:=\underset{x}{\max }?{\lambda }_1(\mathrm{\Omega }?(x+D))$

is large enough in terms of ${\lambda }_1(\mathrm{\Omega })$, then all maximizer sets $x+D$ of ${\mu }_{\mathrm{\Omega }}$ are close to each maximum point $x_0$ of ${?}_{{\lambda }_1}$.Second, we discuss the distribution of ${?}_{{\lambda }_1(\mathrm{\Omega })}$ and the possibility to inscribe wavelength balls at a given point in Ω.Finally, we specify our observations to convex obstacles D and show that if ${\mu }_{\mathrm{\Omega }}$ is sufficiently large with respect to ${\lambda }_1(\mathrm{\Omega })$, then all maximizers $x+D$ of ${\mu }_{\mathrm{\Omega }}$ contain all maximum points $x_0$ of ${?}_{{\lambda }_1(\mathrm{\Omega })}$.     

Degree sequence conditions for maximally edge-connected and super-edge-connected digraphs depending on the clique number

Let $D$ be a finite and simple digraph with vertex set $V\left(D\right)$. For a vertex $v\in V\left(D\right)$, the degree $d\left(v\right)$ of $v$ is defined as the minimum value of its out-degree $d^{+}\left(v\right)$ and its in-degree $d^{?}\left(v\right)$. If $D$ is a graph or a digraph with minimum degree $\delta$ and edge-connectivity $\lambda$, then $\lambda \le \delta$. A graph or a digraph is maximally edge-connected if $\lambda =\delta$. A graph or a digraph is called super-edge-connected if every minimum edge-cut consists of edges adjacent to or from a vertex of minimum degree.In this note we present degree sequence conditions for maximally edge-connected and super-edge-connected digraphs depending on the clique number of the underlying graph.