Opuscula Mathematica
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ISSN 1232-9274
e-ISSN: 2300-6919
Issue Date
2010
Volume
Vol. 30
Number
No. 2
Description
Journal Volume
Opuscula Mathematica
Vol. 30 (2010)
Projects
Pages
Articles
On chromatic equivalence of a pair of K4-homeomorphs
(Wydawnictwa AGH, 2010) Catada-Ghimire, S.; Roslan, H.; Peng, Yee-hock
Let $P(G, \lambda)$ be the chromatic polynomial of a graph $G$. Two graphs $G$ and $H$ are said to be chromatically equivalent, denoted $G∼H$, if $PP(G, \lambda)=P(H, \lambda)$. We write $[G] = \{H| H \sim G\}$. If $[G] = \{G\}$, then $G$ is said to be chromatically unique. In this paper, we discuss a chromatically equivalent pair of graphs in one family of $K_{4}$-homeomorphs, $K_{4}(1,2,8,d,e,f)$. The obtained result can be extended in the study of chromatic equivalence classes of $K_{4}(1,2,8,d,e,f)$ and chromatic uniqueness of $K_{4}$-homeomorphs with girth $11$.
Pseudospectral method for semilinear partial functional differential equations
(Wydawnictwa AGH, 2010) Czernous, Wojciech
We present a convergence result for two spectral methods applied to an initial boundary value problem with functional dependence of Volterra type. Explicit condition of Courant-Friedrichs-Levy type is assumed on time step τ and the number $N$ of collocation points. Stability statements and error estimates are written using continuous norms in weighted Jacobi spaces.
On some families of arbitrarily vertex decomposable spiders
(Wydawnictwa AGH, 2010) Juszczyk, Tomasz; Zioło, Irmina Anna
A graph $G$ of order n is called arbitrarily vertex decomposable if for each sequence $(n_{1}...,n_{k})$ of positive integers such that $\Sigma^{k}_{i=1}n_{i}=n$, there exists a partition $(V_{1},...,V_{k})$ of the vertex set of G such that for every $i\in{1,....,k}$ the set $V_{i}$ induces a connected subgraph of $G$ on $n_{i}$ vertices. A spider is a tree with one vertex of degree at least 3. We characterize two families of arbitrarily vertex decomposable spiders which are homeomorphic to stars with at most four hanging edges.
Fréchet differential of a power series in Banach algebras
(Wydawnictwa AGH, 2010) Silvestri, Benedetto
We present two new forms in which the Fréchet differential of a power series in a unitary Banach algebra can be expressed in terms of absolutely convergent series involving the commutant $C(T) : A \mapsto [A,T]$. Then we apply the results to study series of vector-valued functions on domains in Banach spaces and to the analytic functional calculus in a complex Banach space.
Domination hypergraphs of certain digraphs
(Wydawnictwa AGH, 2010) Sonntag, Martin; Teichert, Hanns-Martin
If $D=(V,A)$ is a digraph, its domination hypergraph $\mathcal{DH}(D) = (V,\mathcal{E})$ has the vertex set $V$ and $e \subseteq V$ is an edge of $\mathcal{DH}(D)$ if and only if e is a minimal dominating set of $D$. We investigate domination hypergraphs of special classes of digraphs, namely tournaments, paths and cycles. Finally, using a special decomposition/composition method we construct edge sets of domination hypergraphs of certain digraphs.