Let $G$ be a connected graph. Gao et al. first introduced the invariant of $G$: $\Delta M(G)=M_{o}(G)-irr(G)$, and raised a problem: how to determine the extremal graphs among all connected graphs of order $n$ with respect to $\Delta M(G)$, where $M_{o}(G)$ and $irr(G)$ stand for the Mostar index and irregularity of $G$, respectively. In this paper, we characterize the upper bounds of $\Delta M(G)$ over all unicyclic graphs and bicyclic graphs with diameter 3, and determine the extremal graphs.
In this paper, we prove that if $n\geq4$ and $a\ge 0$ are integers satisfying $a<\frac{n}{3}$, then
$$\gcd\left(\left\{\binom{n}{k}:a<k<n-a\right\}\right)=\prod_{n=p^{m}+b(n,p),\ 0\le b(n,p)\leq a,} p,$$ where $\binom{n}{k}=\frac{n!}{k!(n-k)!}$, and the product in the right hand side runs through all primes $p$ such that $ n=p^{m}+b(n,p), m\in\mathbb{N}$ and $0\le b(n,p)\leq a$.
As an application of our result, we give an answer to a problem in Hong [16].
In this paper a SIR model for the parallel propagation of two new coronaviruses with time delay is studied. Firstly, the existence and uniqueness of the global positive solution of the system is proved. Secondly, the sufficient conditions for persistence and extinction of the new coronavirus and the variant new coronavirus are obtained by constructing a appropriate Lyapunov function. Finally, the relevant conclusions are verified by numerical simulations.
For improving the application performance of Logistic regression models on classification problems, this paper develops a double adaptive elastic net by combining the adaptive Lasso and adaptive Ridge. The double adaptive elastic net has both the oracle property and the adaptive grouping effect, which ensures that it can effectively estimate parameters and accurately select important variables under certain assumed premises and consequently, makes the established Logistic regression model simple and precise. Simulation and case analysis show that the double adaptive elastic net is suitable for medium or high correlation cases with adaptive grouping effect, and its performance of improving Logistic regression is equal to or better than that of the elastic net and other partial improvement methods.
This paper is devoted to the nonparametric goodness-of-fit test for integrated diffusion processes. Firstly, a nonparametric test is constructed for testing whether the drift function of a integrated diffusion process is of a known parametric form with unknown parameters. Secondly, a test statistic for goodness-of-fit test is obtained by applying the empirical likelihood technique. And finally, the asymptotic distribution of the test statistic is established, and then the proposed test method is applied to an example to verify its effectiveness.
In the resource scheduling network, the availability of resource scheduling is equivalent to the existence of the fractional factor in the corresponding network graph. The study on the existence of fractional factors in specific graph structure can help engineers design and construct the network with efficient use of resources. A graph $G$ is called an all fractional $(g,f,n',m)$-critical deleted graph if after removing any $n'$ vertices from $G$ the remaining graph is still an all fractional $(g,f,m)$-deleted graph. In this paper, we present two binding number conditions for a graph to be an all fractional $(g,f,n',m)$-critical deleted graph, and illustrate the results are sharp with examples.
For a simple graph $G$ with edge set $E(G)$, the exponential inverse forgotten index of $G$ is defined as ~$e^{\frac{1}{\mathcal{F}}}(G)=\sum_{uv\in E(G)}e^{\left(\frac{1}{{d_G^2(u)}}+\frac{1}{{d_G^2(v)}}\right)}$, where $d_G(u)$ is the degree of the vertex $u$ in $G$. In this paper, firstly, we give the minimum value of exponential inverse forgotten index of a tree and determine its corresponding extremal graph. Then, we investigate the maximum value of the exponential inverse forgotten index and describe the structural characteristics of the extremal graph.
In this paper, for an odd prime $p$, some $p$-element three-weight linear codes are constructed by defining set, and the complete weight enumerators of those codes are determined by using Weil sums over the finite field $\mathbb{F}_p$. Furthermore, it is proved that those codes are minimal under certain conditions, and thus suitable for secret sharing schemes.
Especially, a class of those codes with parameters $[p^2-1,3,p^2-p-1]$ are obtained, which are optimal with respect to the Griesmer bound. Our results can be regarded as improvements to some results of Jian et al. in [1].
In this paper, two new classes of primitive Evans triangles are constructed by using
the positive integer solutions of the quadratic equation \ $kx^2-ly^2=2$,
and the trilateral forms and corresponding Evans ratios of these kinds of Evans triangles are given.
