Edge-fault-tolerant pancyclicity and bipancyclicity of Cartesian product graphs with faulty edges

Let r≥ 4 be an even integer. Graph G is r-bipancyclic if it contains a cycle of every even length from r to $2\lfloor \frac{n(G)}{2}\rfloor$, where $n(G)$ is the number of vertices in G. A graph G is r-pancyclic if it contains a cycle of every length from r to $n(G)$, where $r\ge 3$. A graph is k-edge-fault Hamiltonian if, after deleting arbitrary k edges from the graph, the resulting graph remains Hamiltonian. The terms k-edge-fault r-bipancyclic and k-edge-fault r-pancyclic can be defined similarly. Given two graphs G and H, where $n(G)$, $n(H)\ge$ 9, let $k_1$, $k_2\ge 5$ be the minimum degrees of G and H, respectively. This study determined the edge-fault r-bipancyclic and edge-fault r-pancyclic of Cartesian product graph $G\times H$ with some conditions. These results were then used to evaluate the edge-fault pancyclicity (bipancyclicity) of ${\mathit{NQ}}_{m_r,\dots ,m_1}$ and ${\mathit{GQ}}_{m_r,\dots ,m_1}$.     

Suslin’s algorithms for reduction of unimodular rows

A well-known lemma of Suslin says that for a commutative ring $A$ if $(v_1\left(X\right),\dots ,v_n\left(X\right))\in {\left(A\left[X\right]\right)}^n$ is unimodular where $v_1$ is monic and $n\ge 3$, then there exist ${\gamma }_1,\dots ,{\gamma }_{\ell }\in E_{n-1}\left(A\left[X\right]\right)$ such that the ideal generated by $\mathrm{Res}(v_1,e_1.{\gamma }_1{}^t(v_2,\dots ,v_n)),\dots ,\mathrm{Res}(v_1,e_1.{\gamma }_{\ell }{}^t(v_2,\dots ,v_n))$ equals $A$. This lemma played a central role in the resolution of Serre’s Conjecture. In the case where $A$ contains a set $E$ of cardinality greater than $deg{v}_1+1$ such that $y-y^{\prime }$ is invertible for each $y\ne y^{\prime }$ in $E$, we prove that the ${\gamma }_i$ can simply correspond to the elementary operations $L_1\to L_1+y_i{\sum }_{j=2}^{n-1}{u}_{j+1}{L}_j$, $1\le i\le \ell =deg{v}_1+1$, where $u_1{v}_1+\cdots +u_n{v}_n=1$. These efficient elementary operations enable us to give new and simple algorithms for reducing unimodular rows with entries in $K[X_1,\dots ,X_k]$ to ${}^t(1,0,\dots ,0)$ using elementary operations in the case where $K$ is an infinite field. Another feature of this paper is that it shows that the concrete local–global principles can produce competitive complexity bounds.     

Convergence for a family of discrete advection–reaction operators

We define a family of discrete Advection–reaction operators, denoted by $A_{a\lambda }$, which associate to a given scalar sequence $s=\left\{s_n\right\}$ the sequence given by $A_{a\lambda }\left(s\right)\equiv \left\{b_n\right\}$, where $b_n=a_{n?2}{s}_{n?1}+{\lambda }_n{s}_n$ for $n=1,2,\dots$. For $A_{a\lambda }$ we explicitly find their iterates and study their convergence properties. Finally, we show the relationship between the family of discrete operators with the continuous one dimensional advection–reaction equation.     

The Laplacian spectral radius of bicyclic graphs with a given girth

Let $?(n,g)$ be the class of bicyclic graphs on $n$ vertices with girth $g$. Let ${?}_1(n,g)$ be the subclass of $?(n,g)$ consisting of all bicyclic graphs with two edge-disjoint cycles and ${?}_2(n,g)=?(n,g)?{?}_1(n,g)$. This paper determines the unique graph with the maximal Laplacian spectral radius among all graphs in ${?}_1(n,g)$ and ${?}_2(n,g)$, respectively. Furthermore, the upper bound of the Laplacian spectral radius and the extremal graph for $?(n,g)$ are also obtained.     

A new approach toward stabilization in a two-species chemotaxis model with logistic source

This paper aims at providing an alternative approach to study global dynamic properties for a two-species chemotaxis model, with the main novelty being that both populations mutually compete with the other on account of the Lotka–Volterra dynamics. More precisely, we consider the following Neumann initial–boundary value problem

in a bounded domain $\Omega ?{\mathbb{R}}^n,n\ge 1$, with smooth boundary, where $d_1,d_2,d_3,{\chi }_1,{\chi }_2,{\mu }_1,{\mu }_2,a_1,a_2$ are positive constants.When $a_1\in (0,1)$ and $a_2\in (0,1)$, it is shown that under some explicit largeness assumptions on the logistic growth coefficients ${\mu }_1$ and ${\mu }_2$, the corresponding Neumann initial–boundary value problem possesses a unique global bounded solution which moreover approaches a unique positive homogeneous steady state $(u^1,v^1,w^1)$ of above system in the large time limit. The respective decay rate of this convergence is shown to be exponential.When $a_1\ge 1$ and $a_2\in (0,1)$, if ${\mu }_2$ is suitable large, for all sufficiently regular nonnegative initial data $u_0$ and $v_0$ with $u_0?0$ and $v_0?0$, the globally bounded solution of above system will stabilize toward $(0,1,1)$ as $t\to \infty$ in algebraic.     

Relations between matrix sets generated from linear matrix expressions and their applications

Let $A+BXC$ and $A+BX+YC$ be two linear matrix expressions, and denote by $\{A+BXC\}$ and $\{A+BX+YC\}$ the collections of the two matrix expressions when $X$ and $Y$ run over the corresponding matrix spaces. In this paper, we study relationships between the two matrix sets $\{A_1+B_1{X}_1{C}_1\}$ and $\{A_2+B_2{X}_2{C}_2\}$, as well as the two sets $\{A_1+B_1{X}_1+Y_1{C}_1\}$ and $\{A_2+B_2{X}_2+Y_2{C}_2\}$, by using some rank formulas for matrices. In particular, we give necessary and sufficient conditions for the two matrix set inclusions $\{A_1+B_1{X}_1{C}_1\}?\{A_2+B_2{X}_2{C}_2\}$ and $\{A_1+B_1{X}_1+Y_1{C}_1\}?\{A_2+B_2{X}_2+Y_2{C}_2\}$ to hold. We also use the results obtained to characterize relations of solutions of some linear matrix equations.     

Numerical results on some generalized random Fibonacci sequences

Random Fibonacci sequences are stochastic versions of the classical Fibonacci sequence $f_{n+1}=f_n+f_{n?1}$ for $n>0$, and $f_0=f_1=1$, obtained by randomizing one or both signs on the right side of the defining equation and/or adding a “growth parameter.” These sequences may be viewed as coming from a sequence of products of i.i.d. random matrices and their rate of growth measured by the associated Lyapunov exponent. Following the techniques presented by Embree and Trefethen in their numerical paper Embree and Trefethen (1999) [2], we study the behavior of the Lyapunov exponents as a function of the probability $p$ of choosing $+$ in the sign randomization.     

Trees with minimal Laplacian coefficients

Let $G$ be a simple undirected graph with the characteristic polynomial of its Laplacian matrix $L\left(G\right)$, $P(G,\mu )={\sum }_{k=0}^n{(?1)}^k{c}_k{\mu }^{n?k}$. It is well known that for trees the Laplacian coefficient $c_{n?2}$ is equal to the Wiener index of $G$, while $c_{n?3}$ is equal to the modified hyper-Wiener index of the graph. In this paper, we characterize $n$-vertex trees with given matching number $m$ which simultaneously minimize all Laplacian coefficients. The extremal tree $A(n,m)$ is a spur, obtained from the star graph $S_{n?m+1}$ with $n?m+1$ vertices by attaching a pendant edge to each of certain $m?1$ non-central vertices of $S_{n?m+1}$. In particular, $A(n,m)$ minimizes the Wiener index, the modified hyper-Wiener index and the recently introduced Incidence energy of trees, defined as $IE\left(G\right)={\sum }_{k=0}^n\sqrt{{\mu }_k}$, where ${\mu }_k$ are the eigenvalues of signless Laplacian matrix $Q\left(G\right)=D\left(G\right)+A\left(G\right)$. We introduced a general $\rho$ transformation which decreases all Laplacian coefficients simultaneously. In conclusion, we illustrate on examples of Wiener index and Incidence energy that the opposite problem of simultaneously maximizing all Laplacian coefficients has no solution.     

Subresultants and generic monomial bases

Given $n$ polynomials in $n$ variables of respective degrees $d_1,\dots ,d_n$, and a set of monomials of cardinality $d_1\cdots d_n$, we give an explicit subresultant-based polynomial expression in the coefficients of the input polynomials whose non-vanishing is a necessary and sufficient condition for this set of monomials to be a basis of the ring of polynomials in $n$ variables modulo the ideal generated by the system of polynomials. This approach allows us to clarify the algorithms for the Bézout construction of the resultant.