Opuscula Mathematica
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ISSN 1232-9274
e-ISSN: 2300-6919
Issue Date
2007
Volume
Vol. 27
Number
No. 1
Description
Journal Volume
Opuscula Mathematica
Vol. 27 (2007)
Projects
Pages
Articles
Tree domatic number in graphs
(2007) Chen, Xuegang
Dominating set $S$ in a graph $G$ is a tree dominating set of $G$ if the subgraph induced by $S$ is a tree. The tree domatic number of $G$ is the maximum number of pairwise disjoint tree dominating sets in $V(G)$. First, some exact values of and sharp bounds for the tree domatic number are given. Then, we establish a sharp lower bound for the number of edges in a connected graph of given order and given tree domatic number, and we characterize the extremal graphs. Finally, we show that a tree domatic number of a planar graph is at most $4$ and give a characterization of planar graphs with the tree domatic number $3$.
On Lipschitzian operators of substitution generated by set-valued functions
(2007) Ludew, Jakub Jan
We consider the Nemytskii operator, i.e., the operator of substitution, defined by $(N \phi)(x):=G(x,\phi(x))$, where $G$ is a given multifunction. It is shown that if $N$ maps a Hölder space $H_{\alpha}$ into $H_{\beta}$ and $N$ fulfils the Lipschitz condition then $G(x,y)=A(x,y)+B(x), (1)$ where $A(x,\cdot)$ is linear and $A(\cdot ,y),\, B \in H_{\beta}$. Moreover, some conditions are given under which the Nemytskii operator generated by $(1)$ maps $H_{\alpha}$ into $H_{\beta}$ and is Lipschitzian.
Construction of algebraic-analytic discrete approximations for linear and nonlinear hyperbolic equations in R². Part 1
(2007) Luśtyk, Mirosław; Prytula, Mykola
An algebraic-analytic method for constructing discrete approximations of linear hyperbolic equations based on a generalized d'Alembert formula of the Lytvyn and Riemann expressions for Cauchy data is proposed. The problem is reduced to some special case of the fixed point problem.
Approximation of eigenvalues of some unbounded self-adjoint discrete Jacobi matrices by eigenvalues of finite submatrices
(2007) Malejki, Maria
We investigate the problem of approximation of eigenvalues of some self-adjoint operator in the Hilbert space $l^2(\mathbb{N})$ by eigenvalues of suitably chosen principal finite submatrices of an infinite Jacobi matrix that defines the operator considered. We assume the Jacobi operator is bounded from below with compact resolvent. In our research we estimate the asymptotics (with $n\to \infty$) of the joint error of approximation for the first n eigenvalues and eigenvectors of the operator by the eigenvalues and eigenvectors of the finite submatrix of order $n \times n$. The method applied in our research is based on the Rayleigh-Ritz method and Volkmer's results included in . We extend the method to cover a class of infinite symmetric Jacobi matrices with three diagonals satisfying some polynomial growth estimates.
k-perfect geodominating sets in graphs
(2007) Mojdeh, Doost Ali; Rad, Nader Jafari
A perfect geodominating set in a graph $G$ is a geodominating set $S$ such that any vertex $v \in V(G)\setminus S$ is geodominated by exactly one pair of vertices of $S$. A $k$-perfect geodominating set is a geodominating set $S$ such that any vertex $v \in V(G)\setminus S$ is geodominated by exactly one pair $x$, $y$ of vertices of $S$ with $d(x, y) = k$. We study perfect and $k$-perfect geodomination numbers of a graph $G$.