Examples and Counterexamples最新文献

It was conjectured by Asmiati (2018) that the generalized Petersen graph $P\left(n,k\right)$ has a locating chromatic number 4 if and only if $(noddandk=1)$ or $(n=4andk=2)$. In this paper, we give a negative answer to the conjecture posed by Asmiati. As a consequence, we are able to exhibit many counterexamples to the recent conjecture proposed, by proving that if $(5\le n\le 12)$ and $(2\le k\le \lfloor \frac{n-1}{2}\rfloor )$ and $(n,k)\ne (12,5)$, then ${\chi }_{{}_L}\left(P(n,k)\right)=4$.

A code $C\subseteq {\{0,1,2\}}^n$ of length $n$ is called trifferent if for any three distinct elements of $C$ there exists a coordinate in which they all differ. By $T\left(n\right)$ we denote the maximum cardinality of trifferent codes with length $n$. The values $T\left(5\right)=10$ and $T\left(6\right)=13$ were recently determined (Fiore et al., 2022). Here we determine $T\left(7\right)=16$, $T\left(8\right)=20$, and $T\left(9\right)=27$. For the latter case $n=9$ there also exist linear codes attaining the maximum possible cardinality 27.

The research introduces the Modified Sumudu Decomposition Method (MSDM) as a novel approach for solving non-linear ordinary differential equations. Stemming from the Sumudu Transformation (ST), proposed by Watugala in the 1990s, MSDM demonstrates its efficacy through the solution of a specific third-order non-homogeneous nonlinear ordinary differential equation. This method is particularly highlighted for its application in fluid mechanics, specifically addressing a boundary layer problem. Furthermore, the study employs Pade´ Approximants to evaluate a crucial parameter, ρ=φ''(0), and compares the results with other established methods, including Modified Laplace Decomposition Method (MLDM), Modified Adomian Decomposition Method (MADM), Modified Variational Iteration Method (MVIM), and the Homotopy Perturbation Method (HPM). The findings not only contribute to the advancement of mathematical techniques for solving complex differential equations but also provide a comparative analysis, elucidating the strengths and limitations of different methodologies. This research is anticipated to have significant implications for researchers and practitioners in the field, offering a valuable toolkit for tackling a wide range of mathematical modeling challenges.

As proved by Marchenkov and Mazzanti, every Kalmar function can be represented by arithmetic terms. We display one of such terms to represent the factorial function, and as a consequence, we get an example of an arithmetic term which represents a function whose image is the set of primes.

A recent classification of flag-transitive 2-designs with parameters $(v,k,\lambda )$ whose replication number $r$ is coprime to $\lambda$ gives rise to eight possible infinite families of 2-designs, some of which are with new parameters. In this note, we give explicit constructions for two of these families of 2-designs, and show that for a given positive integer $q=3^{2n+1}⩾27$, there exist 2-designs with parameters $(q^3+1,q^i,q^i-1)$, for $i=1,2$, admitting the Ree group ${}^2{G}_2\left(q\right)$ as their automorphism groups.

In this note we show that for each positive integer $a⩾2$ there exist infinitely many trees whose spectral radius is equal to $\sqrt{2a}$. Such trees are obtained by replacing the central edge of the double star $S(a,2a-2)$ with suitable bidegreed caterpillars.

In this corrigendum, we correct some errors in the proof of Theorem 2.1 in “On the conjecture of Sombor energy of a graph” [Examples and Counterexamples 3 (2023) 100115].

In this note it is shown that there is a bounded linear operator $T$ on the Hardy Hilbert space $H^2$ and a vector $f$ in $H^2$ such that the closure of the set $\{\alpha T^{n}f:\alpha \in ℂ,n\ge 0\}$ is not $H^2$, but for every subsequence ${\left(n_k\right)}_{k=1}^{\infty }$ the closed linear span of $\{T^{n_k}f:k\ge 1\}$ is the whole space $H^2$. Furthermore, the closure of $\{T^{n}g:n\ge 0\}$ is $H^2$ for some $g\in H^2$.

We termed the Pál type interpolation problem as PTIP. Here we studied the regularity of $(0,1)$-PTIP and $(0,2)$-PTIP, where we omitted a real or complex node from a set of zeros of polynomials with complex coefficients.

In this paper, we give a main example indicating the ineffectiveness of the local fractional derivatives on the Riemann curvature tensor that is a common tool in calculating curvature of a Riemannian manifold. For this, first we introduce a general local fractional derivative operator that involves the mostly used ones in the literature as conformable, alternative, truncated $M-$ and $V-$fractional derivatives. Then, according to this general operator, a particular Riemannian metric on the real affine space $R^n$ that is different from the Euclidean one is defined. In conclusion, our main example states that the Riemann curvature tensor of $R^n$ endowed with this particular metric is identically 0, that is, one is locally isometric to Euclidean space.