bluuebunny/crossref_metadata_2025_split huggingface dataset:
PENGARUH PERANAN DAN POLA ASUH ORANG TUA TERHADAP HASIL BELAJAR SISWA SEKOLAH DASAR NEGERI (SDN) DI KECAMATAN BOGOR BARAT
Authors: Ratna Ningrum Wulan 
Published: January 2020

The aims of the study is to analyze the partial and simultaneous influence of parent role and parenting patterns on students learning achievement at Elementary Public School (Sekolah Dasar Negeri) in the District of West Bogor. A survey was conducted by distributing questionnaires to the respondents, complemented with interview for qualitative data. Student achievement is collected as part of the research data. The respondents are 100 parents from 50 schools throughout the district. The results of the study reveals that there is a positive and significant partial influence of parent role and parenting pattern on student learning achievement. Parent role and parenting pattern also have positive and significant simultaneous effect on student learning achievement. It can be concluded that efforts to improve student learning achievement should take into account parent role and parenting patterns.

Tujuan dari penelitian ini adalah untuk menganalisis pengaruh parsial dan simultan peran orangtua dan pola pengasuhan terhadap prestasi belajar siswa di Sekolah Dasar Negeri di Kabupaten Bogor Barat. Pengumpulan data dilakukan melalui survei dengan menyebarkan kuesioner kepada responden, dilengkapi dengan wawancara untuk data kualitatif. Prestasi siswa dikumpulkan sebagai bagian dari data penelitian. Responden penelitian adalah 100 orang tua dari 50 sekolah di seluruh kabupaten. Hasil penelitian menunjukkan bahwa terdapat pengaruh parsial yang positif dan signifikan peran orangtua dan pola pengasuhan terhadap prestasi belajar siswa. Peran orang tua dan pola pengasuhan secara simultan juga berpengaruh positif dan signifikan terhadap prestasi belajar siswa. Dengan demikian dapat disimpulkan bahwa upaya peningkatan prestasi belajar siswa harus memperhatikan peran orang tua dan pola pengasuhan siswa oleh orang tua di rumah.
Intensitas Penggunaan e-filling Wajib Pajak Orang Pribadi Yang Dipengaruhi Persepsi Kemudahan Pengguna e filling” pada Wajib Pajak Orang Pribadi( Studi Kasus Pada Instansi Swasta Di Bandung)
Authors: Astuti Wati Aris , Febriansyah` Angky 
Published: January 2020

Penggunaan e-filing merupakan suatu proses di mana wajib pajak orang pribadi menggunakan sistem e-filing untuk melaporkan SPT secara online. E-filing digunakan untuk memberi keuntungan dan kemudahan bagi pihak Direktorat Jenderal Pajak dan wajib pajak orang pribadi dalam pelaporan SPT (Surat Pemberitahuan). Dengan adanya e-filing wajib pajak akan mendapatkan keuntungan lebih efisiensi dan efektivitas dalam membuat pelaporan SPT tahunan tanpa perlu mengkhawatirkan jam kerja operasional kantor pajak karena wajib pajak orang pribadi dapat menggunakan e-filing tanpa perlu datang ke kantor pajak.
Dengan e-filling maka diharapkan pelaporan pajak dapat dilakukan dengan cepat, mudah, dan aman. SPT yang dikirimkan akan di enkripsi sehingga terjamin kerahasiaannya. Pihak-pihak yang tidak berkepentingan tidak akan dapat mengetahui isi dari SPT tersebut.Hasil dari penelitian ini bahwa Persepsi Kemudahan Pengguna e filling” pada Wajib Pajak Orang Pribadi berpengaruh terhadap Intensitas Penggunaan e-filling Wajib Pajak Orang Pribadi.

Kunci : e-filing, SPT, Wajib Pajak
PENTINGNYA PELATIHAN DAN PENGEMBANGAN SUMBER DAYA MANUSIA TERHADAP PENINGKATAN KINERJA KARYAWAN PADA KFC JEMBER
Authors: Ashary Luckman 
Published: January 2020

Tujuan penelitian ini adalah untuk mengetahui peran dan evaluasi manajer SDM KFC Jember dalam meningkatkan kinerja karyawan serta mengetahui metode pelatihan dan pengembangan yang diterapkan bagi karyawan KFC Jember kendala- yang dihadapi Manajer KFC Jember dalam melakukan peningkatan kinerja karyawan. Jenis penelitian ini menggunakan pendekatan studi kasus dengan metode deskriptif kualitatif. Lokasi Penelitian dilakukan pada karyawan KFC Jember yang beralamat di Jalan Gajah Mada No.74, Jember Kidul, Kaliwates, Kabupaten Jember, Jawa Timur. Theoretical sampling yang digunakan pada penelitian ini adalah purposive sampling. Hasil penelitian menyebutkan metode pelatihan calon karyawan yang digunakan KFC Jember adalah Lecture dan on the job training. Metode pengembangan karyawan yang digunakan KFC Jember adalah job rotasi lalu dengan memakai kembali metode pelatihan (on the job training) dan understudy. Pelatihan calon karyawan dan pengembangan karyawan yang dilakukan telah sesuai dengan prosedur yang telah ditetapkan oleh perusahaan.
Експериментальне дослідження стоку з водопроникного бетонного покриття для високоінтенсивних дощів малої тривалості
Authors: Zhuk V. M. , Kachmar I. Z. , Fasuliak V. Ye. 
Published: January 2020

Використання водопроникних удосконалених покриттів (ВУП) – ефективний метод управління дощовим стоком на урбанізованих територіях. Основним гідравлічним параметром під час моделювання систем дощового водовідведення є гідрографи стоку для дощів розрахункової тривалості та інтенсивності. Регулювання дощового стоку за допомогою ВУП на сьогодні достатньо добре вивчене, проте все ще є нез'ясовані питання щодо функції "дощові опади – стік", особливо для короткотривалих дощів високої інтенсивності. Виконано серію експериментальних досліджень дощового стоку з фрагмента натурного ВУП на підставі пористого бетону. Використано типову конструкцію ВУП з нижнім шаром зі щебеню товщиною 220 мм та верхнім шаром із водопроникного бетону товщиною 100 мм. Розміри дослідного ВУП – 3,0×0,3×0,32 м, поздовжній похил покриття – 0,01. Регулювальні властивості ВУП отримано для модельних дощів особливо високої інтенсивності 10300 л/(с·га) з тривалістю від 15 с до 60 с. Дощ тривалістю tcon.0 = 38,4 с відповідає часу поверхневої концентрації для аналогічного повністю водонепроникного асфальтового покриття. З'ясовано, що використання ВУП зменшує у такому випадку максимальну витрату стоку в 3,1 раза – від 0,927 л/с до 0,302 л/с, тоді як час досягнення максимальної витрати зростає на 20,3 %. Час концентрації стоку tcon.ВУП = 112,0 с з дослідженого ВУП у 2,92 раза більший порівняно з аналогічним водонепроникним басейном.
Values of Domination Numbers of the Queen's Graph
Authors: Östergård Patric R. J. , Weakley William D. 
Published: January 2020

The queen's graph $Q_{n}$ has the squares of the $n \times n$ chessboard as its vertices; two squares are adjacent if they are in the same row, column, or diagonal. Let $\gamma (Q_{n})$ and $i(Q_{n})$ be the minimum sizes of a dominating set and an independent dominating set of $Q_{n}$, respectively. Recent results, the Parallelogram Law, and a search algorithm adapted from Knuth are used to find dominating sets. New values and bounds:(A) $\gamma (Q_n) = \lceil n/2 \rceil$ is shown for 17 values of $n$ (in particular, the set of values for which the conjecture $\gamma (Q_{4k+1}) = 2k + 1$ is known to hold is extended to $k \leq 32$);(B) $i(Q_n) = \lceil n/2 \rceil$ is shown for 11 values of $n$, including 5 of those from (A);(C) One or both of $\gamma (Q_n)$ and $i(Q_n)$ is shown to lie in $\{ \lceil n/2 \rceil $, $\lceil n/2 \rceil + 1 \}$ for 85 values of $n$ distinct from those in (A) and (B).Combined with previously published work, these results imply that for $n \leq 120$, each of $\gamma (Q_n)$ and $i(Q_n)$ is either known, or known to have one of two values.Also, the general bounds $\gamma (Q_n) \leq 69n/133 + O(1)$ and $i(Q_n) \leq 61n/111 + O(1)$ are established. Comment added August 25th 2003.Corrigendum added October 5th 2017.
Anti-Codes in Terms of Berlekamp's Switching Game
Authors: Schauz Uwe 
Published: January 2020

We view a linear code (subspace) $C\leq\mathbb{F}_{q}^n$ as a light pattern on the \(n\)-dimensional Berlekamp Board $\mathbb{F}_{q}^n$ with $q^n$ light bulbs. The lights corresponding to elements of $C$ are ON, the others are OFF. Then we allow axis-parallel switches of complete rows, columns, etc. We show that the dual code $C^\perp$ contains a vector $v$ of full weight, i.e. $v_1,v_2,\dots,v_n\neq0$, if and only if the light pattern $C$ cannot be switched off. Generalizations of this allow us to describe anti-codes with maximal weight $\delta$ in a similar way, or, alternatively, in terms of a switching game in projective space. We provide convenient bases and normal forms to the modules of all light patterns of the generalized games. All our proofs are purely combinatorial and simpler than the algebraic ones used for similar results about anti-codes in $\mathbb{Z}_k^n$. Aside from coding theory, the game is also of interest in the study of nowhere-zero points of matrices and nowhere-zero flows and colorings of graphs.
Properties of $\theta$-super positive graphs
Authors: Ku Cheng Yeaw , Wong Kok Bin 
Published: January 2020

Let the matching polynomial of a graph $G$ be denoted by $\mu (G,x)$. A graph $G$ is said to be $\theta$-super positive if $\mu(G,\theta)\neq 0$ and $\mu(G\setminus v,\theta)=0$ for all $v\in V(G)$. In particular, $G$ is $0$-super positive if and only if $G$ has a perfect matching. While much is known about $0$-super positive graphs, almost nothing is known about $\theta$-super positive graphs for $\theta \neq 0$. This motivates us to investigate the structure of $\theta$-super positive graphs in this paper. Though a $0$-super positive graph need not contain any cycle, we show that a $\theta$-super positive graph with $\theta \neq 0$ must contain a cycle. We introduce two important types of $\theta$-super positive graphs, namely $\theta$-elementary and $\theta$-base graphs. One of our main results is that any $\theta$-super positive graph $G$ can be constructed by adding certain type of edges to a disjoint union of $\theta$-base graphs; moreover, these $\theta$-base graphs are uniquely determined by $G$. We also give a characterization of $\theta$-elementary graphs: a graph $G$ is $\theta$-elementary if and only if the set of all its $\theta$-barrier sets form a partition of $V(G)$. Here, $\theta$-elementary graphs and $\theta$-barrier sets can be regarded as $\theta$-analogue of elementary graphs and Tutte sets in classical matching theory.
Two-Part Set Systems
Authors: Erdős Péter L. , Gerbner Dániel , Lemons Nathan , Mubayi Dhruv , Palmer Cory , Patkós Balázs 
Published: January 2020

The two part Sperner theorem of Katona and Kleitman states that if $X$ is an $n$-element set with partition $X_1 \cup X_2$, and $\mathcal{F}$ is a family of subsets of $X$ such that no two sets $A, B \in \mathcal{F}$ satisfy $A \subset B$ (or $B \subset A$) and $A \cap X_i=B\cap X_i$ for some $i$, then $|\mathcal{F}| \le {n \choose \lfloor n/2\rfloor}$. We consider variations of this problem by replacing the Sperner property with the intersection property and considering families that satisfy various combinations of these properties on one or both parts $X_1$, $X_2$. Along the way, we prove the following new result which may be of independent interest: let $\mathcal{F},\mathcal{G}$ be intersecting families of subsets of an $n$-element set that are additionally cross-Sperner, meaning that if $A \in\mathcal{F}$ and $B \in \mathcal{G}$, then $A \not\subset B$ and $B \not\subset A$. Then $|\mathcal{F}| +|\mathcal{G}| \le 2^{n-1}$ and there are exponentially many examples showing that this bound is tight.
On the Number of Indecomposable Permutations with a Given Number of Cycles
Authors: Cori Robert , Mathieu Claire , Robson John Michael 
Published: January 2020

A permutation $a_1a_2\ldots a_n$ is indecomposable if there does not exist $p<n$ such that $a_1a_2\ldots a_p$ is a permutation of $\{ 1,2,\ldots,p\}$. We consider the probability that a permutation of ${\mathbb S}_n$ with $m$ cycles is indecomposable and prove that this probability is monotone non-increasing in $n$.We compute also the asymptotic probability when $n$ goes to infinity with $m/n$ tending to a fixed ratio. The asymptotic probability is monotone in $m/n$, and there is no threshold phenomenon: it degrades gracefully from 1 to 0. When $n=2m$, a slight majority ($51.117\ldots$ percent) of the permutations are indecomposable.
Partitions with Distinct Multiplicities of Parts: On An "Unsolved Problem" Posed By Herbert Wilf
Authors: Fill James Allen , Janson Svante , Ward Mark Daniel 
Published: January 2020

Wilf's Sixth Unsolved Problem asks for any interesting properties of the set of partitions of integers for which the (nonzero) multiplicities of the parts are all different. We refer to these as Wilf partitions. Using $f(n)$ to denote the number of Wilf partitions, we establish lead-order asymptotics for $\ln{f(n)}$.
Combinatorial Interpretations of Particular Evaluations of Complete and Elementary Symmetric Functions
Authors: Mongelli Pietro 
Published: January 2020

The Jacobi-Stirling numbers and the Legendre-Stirling numbers of the first and second kind were first introduced by Everitt et al. (2002) and (2007) in the spectral theory. In this paper we note that Jacobi-Stirling numbers and Legendre-Stirling numbers are specializations of elementary and complete symmetric functions. We then study combinatorial interpretations of this specialization and obtain new combinatorial interpretations of the Jacobi-Stirling and Legendre-Stirling numbers.
Maximum Frustration in Bipartite Signed Graphs
Authors: Bowlin Garry S 
Published: January 2020

A signed graph is a graph where each edge is labeled as either positive or negative. A circle is positive if the product of edge labels is positive. The frustration index is the least number of edges that need to be removed so that every remaining circle is positive. The maximum frustration of a graph is the maximum frustration index over all possible sign labellings. We prove two results about the maximum frustration of a complete bipartite graph $K_{l,r}$, with $l$ left vertices and $r$ right vertices. First, it is bounded above by\[ \frac{lr}{2}\left(1-\frac{1}{2^{l-1}}\binom{l-1}{\lfloor \frac{l-1}{2}\rfloor}\right).\] Second, there is a unique family of signed $K_{l,r}$ that reach this bound. Using this fact, exact formulas for the maximum frustration of $K_{l,r}$ are found for $l \leq 7$.
Finite Homomorphism-Homogeneous Permutations via Edge Colourings of Chains
Authors: Dolinka Igor , Jungábel Éva 
Published: January 2020

A relational structure is homomorphism-homogeneous if any homomorphism between its finite substructures extends to an endomorphism of the structure in question. In this note, we characterise all permutations on a finite set enjoying this property. To accomplish this, we switch from the more traditional view of a permutation as a set endowed with two linear orders to a different representation by a single linear order (considered as a directed graph with loops) whose non-loop edges are coloured in two colours, thereby `splitting' the linear order into two posets.
Inequivalence of Difference Sets: On a Remark of Baumert
Authors: Ó Catháin Padraig 
Published: January 2020

An often cited statement of Baumert in his book Cyclic difference sets asserts that four well known families of cyclic $(4t-1,2t-1,t-1)$ difference sets are inequivalent, apart from a small number of exceptions with $t< 8$. We are not aware of a proof of this statement in the literature.Three of the families discussed by Baumert have analogous constructions in non-cyclic groups. We extend his inequivalence statement to a general inequivalence result, for which we provide a complete and self-contained proof. We preface our proof with a survey of the four families of difference sets, since there seems to be some confusion in the literature between the cyclic and non-cyclic cases.
Existence of 2-$(v,k,1)$ Designs Admitting a Block-Transitive Group of Affine Type
Authors: Shifeng Ding , Weijun Liu 
Published: January 2020

In this paper we use Weil's estimate on character sums to prove that large admissible prime powers $q$ admit 2-$(q,k,1)$ designs having block-transitive automorphism groups in AGL$(1, q)$.
Ramsey Properties of Random Subgraphs of Pseudo-Random Graphs
Authors: Shen Jia 
Published: January 2020

Let $G=(V,E)$ be a $d$-regular graph of order $n$. Let $G_p$ be the random subgraph of $G$ for which each edge is selected from $E(G)$ independently at random with probability $p$. For a fixed graph $H$, define $m(H):=$max$\{e(H')/(v(H')-1):H' \subseteq H\}$. We prove that $n^{(m(H)-1)/m(H)}/d$ is a threshold function for $G_p$ to satisfy Ramsey, induced Ramsey, and canonical Ramsey properties with respect to vertex coloring, respectively, provided the eigenvalue $\lambda$ of $G$ that is second largest in absolute value is significantly smaller than $d$.As a consequence, it is also shown that $\displaystyle n^{(m(H)-1)/m(H)}/d$ is a threshold function for $G_p$ to contain a family of vertex disjoint copies of $H$ (an $H$ packing) that covers $(1-o(1))n$ vertices of $G$. Using a similar argument, the sharp threshold function for $G_p$ to contain $H$ as a subgraph is obtained as well.
An Ordered Turán Problem for Bipartite Graphs
Authors: Timmons Craig 
Published: January 2020

Let $F$ be a graph. A graph $G$ is $F$-free if it does not contain $F$ as a subgraph. The Turán number of $F$, written $\textrm{ex}(n,F)$, is the maximum number of edges in an $F$-free graph with $n$ vertices. The determination of Turán numbers of bipartite graphs is a challenging and widely investigated problem. In this paper we introduce an ordered version of the Turán problem for bipartite graphs. Let $G$ be a graph with $V(G) = \{1, 2, \dots , n \}$ and view the vertices of $G$ as being ordered in the natural way. A zig-zag $K_{s,t}$, denoted $Z_{s,t}$, is a complete bipartite graph $K_{s,t}$ whose parts $A = \{n_1 < n_2 < \dots < n_s \}$ and $B = \{m_1 < m_2 < \dots < m_t \}$ satisfy the condition $n_s < m_1$. A zig-zag $C_{2k}$ is an even cycle $C_{2k}$ whose vertices in one part precede all of those in the other part. Write $\mathcal{Z}_{2k}$ for the family of zig-zag $2k$-cycles. We investigate the Turán numbers $\textrm{ex}(n,Z_{s,t})$ and $\textrm{ex}(n,\mathcal{Z}_{2k})$. In particular we show $\textrm{ex}(n, Z_{2,2}) \leq \frac{2}{3}n^{3/2} + O(n^{5/4})$. For infinitely many $n$ we construct a $Z_{2,2}$-free $n$-vertex graph with more than $(n - \sqrt{n} - 1) + \textrm{ex} (n,K_{2,2})$ edges.
Graph Cospectrality using Neighborhood Matrices
Authors: Bessa Aline Duarte , Rocha-Neto Ivan Carmo , Pinho Suani Tavares Rubim de , Andrade Roberto Fernandes Silva , Petit Lobao Thierry Correa 
Published: January 2020

In this note we address the problem of graph isomorphism by means of eigenvalue spectra of different matrix representations: the neighborhood matrix $\hat{M}$, its corresponding signless Laplacian $Q_{\hat{M}}$, and the set of higher order adjacency matrices $M_{\ell}$s. We find that, in relation to graphs with at most 10 vertices, $Q_{\hat{M}}$ leads to better results than the signless Laplacian $Q$; besides, when combined with $\hat{M}$, it even surpasses the Godsil and McKay switching method.
The Optimal Drawings of $K_{5,n}$
Authors: Hernández-Vélez César , Medina Carolina , Salazar Gelasio 
Published: January 2020

Zarankiewicz's Conjecture (ZC) states that the crossing number cr$(K_{m,n})$ equals $Z(m,n):=\lfloor{\frac{m}{2}}\rfloor \lfloor{\frac{m-1}{2}}\rfloor \lfloor{\frac{n}{2}}\rfloor \lfloor{\frac{n-1}{2}}\rfloor$. Since Kleitman's verification of ZC for $K_{5,n}$ (from which ZC for $K_{6,n}$ easily follows), very little progress has been made around ZC; the most notable exceptions involve computer-aided results. With the aim of gaining a more profound understanding of this notoriously difficult conjecture, we investigate the optimal (that is, crossing-minimal) drawings of $K_{5,n}$. The widely known natural drawings of $K_{m,n}$ (the so-called Zarankiewicz drawings) with $Z(m,n)$ crossings contain antipodal vertices, that is, pairs of degree-$m$ vertices such that their induced drawing of $K_{m,2}$ has no crossings. Antipodal vertices also play a major role in Kleitman's inductive proof that cr$(K_{5,n}) = Z(5,n)$. We explore in depth the role of antipodal vertices in optimal drawings of $K_{5,n}$, for $n$ even. We prove that if {$n \equiv 2$ (mod $4$)}, then every optimal drawing of $K_{5,n}$ has antipodal vertices. We also exhibit a two-parameter family of optimal drawings $D_{r,s}$ of $K_{5,4(r+s)}$ (for $r,s\ge 0$), with no antipodal vertices, and show that if $n\equiv 0$ (mod $4$), then every optimal drawing of $K_{5,n}$ without antipodal vertices is (vertex rotation) isomorphic to $D_{r,s}$ for some integers $r,s$. As a corollary, we show that if $n$ is even, then every optimal drawing of $K_{5,n}$ is the superimposition of Zarankiewicz drawings with a drawing isomorphic to $D_{r,s}$ for some nonnegative integers $r,s$.
Rainbow Connection of Sparse Random Graphs
Authors: Frieze Alan , Tsourakakis Charalampos E. 
Published: January 2020

An edge colored graph $G$ is rainbow edge connected if any two vertices are connected by a path whose edges have distinct colors. The rainbow connectivity of a connected graph $G$, denoted by $rc(G)$, is the smallest number of colors that are needed in order to make $G$ rainbow connected. In this work we study the rainbow connectivity of binomial random graphs at the connectivity threshold $p=\frac{\log n+\omega}{n}$ where $\omega=\omega(n)\to\infty$ and ${\omega}=o(\log{n})$ and of random $r$-regular graphs where $r \geq 3$ is a fixed integer. Specifically, we prove that the rainbow connectivity $rc(G)$ of $G=G(n,p)$ satisfies $rc(G) \sim \max\{Z_1,\text{diam}(G)\}$ with high probability (whp). Here $Z_1$ is the number of vertices in $G$ whose degree equals 1 and the diameter of $G$ is asymptotically equal to $\frac{\log n}{\log\log n}$ whp. Finally, we prove that the rainbow connectivity $rc(G)$ of the random $r$-regular graph $G=G(n,r)$ whp satisfies $rc(G) =O(\log^{2\theta_r}{n})$ where $\theta_r=\frac{\log (r-1)}{\log (r-2)}$ when $r\geq 4$ and $rc(G) =O(\log^4n)$ whp when $r=3$.
New Computational Upper Bounds for Ramsey Numbers $R(3,k)$
Authors: Goedgebeur Jan , Radziszowski Stanisław P. 
Published: January 2020

Using computational techniques we derive six new upper bounds on the classical two-color Ramsey numbers: $R(3,10) \le 42$, $R(3,11) \le 50$, $R(3,13) \le 68$, $R(3,14) \le 77$, $R(3,15) \le 87$, and $R(3,16) \le 98$. All of them are improvements by one over the previously best known bounds. Let $e(3,k,n)$ denote the minimum number of edges in any triangle-free graph on $n$ vertices without independent sets of order $k$. The new upper bounds on $R(3,k)$ are obtained by completing the computation of the exact values of $e(3,k,n)$ for all $n$ with $k \leq 9$ and for all $n \leq 33$ for $k = 10$, and by establishing new lower bounds on $e(3,k,n)$ for most of the open cases for $10 \le k \le 15$. The enumeration of all graphs witnessing the values of $e(3,k,n)$ is completed for all cases with $k \le 9$. We prove that the known critical graph for $R(3,9)$ on 35 vertices is unique up to isomorphism. For the case of $R(3,10)$, first we establish that $R(3,10)=43$ if and only if $e(3,10,42)=189$, or equivalently, that if $R(3,10)=43$ then every critical graph is regular of degree 9. Then, using computations, we disprove the existence of the latter, and thus show that $R(3,10) \le 42$.
Integer Points in Knapsack Polytopes and $s$-Covering Radius
Authors: Aliev Iskander , Henk Martin , Linke Eva 
Published: January 2020

Given a matrix $A\in \mathbb{Z}^{m\times n}$ satisfying certain regularity assumptions, we consider for a positive integer $s$ the set ${\mathcal F}_s(A)\subset \mathbb{Z}^m$ of all vectors $b\in \mathbb{Z}^m$ such that the associated knapsack polytope\begin{equation*}P(A, b)=\{ x \in \mathbb{R}^n_{\ge 0}: A x= b\}\end{equation*}contains at least $s$ integer points. We present lower and upper bounds on the so called diagonal $s$-Frobenius number associated to the set ${\mathcal F}_s(A)$. In the case $m=1$ we prove an optimal lower bound for the $s$-Frobenius number, which is the largest integer $b$ such that $P(A,b)$ contains less than $s$ integer points.