We consider the class of ▫$I$▫-graphs, which is a generalization of the class of the generalized Petersen graphs. We show that two ▫$I$▫-graphs ▫$I(n, j, k)$▫ and ▫$I(n, j_1, k_1)$▫ are isomorphic if and only if there exists an integer ▫$a$▫ relatively prime to $n$ such that either ▫$\{j_1, k_1\} = \{aj \mod n, \; ak \mod n \}$▫ or ▫$\{j_1, k_1\} = \{aj \mod n, \; -ak \mod n\}$▫. This result has an application in the enumeration of non-isomorphic ▫$I$▫-graphs and unit-distance representations of generalized Petersen graphs.
COBISS.SI-ID: 16069977
In this paper, a classical problem of the construction of a cubic ▫$G^1$▫ continuous interpolatory spline curve is considered. The only data prescribed are interpolation points, while tangent directions are unknown. They are constructed automatically in such a way that a particular minimization of the strain energy of the spline curve is applied. The resulting spline curve is constructed locally and is regular, cusp-, loop- and fold-free. Numerical examples demonstrate that it is satisfactory as far as the shape of the curve is concerned.
COBISS.SI-ID: 15770969
Many different approaches have been proposed for the challenging problem of visually analyzing large networks. Clustering is one of the most promising. In this paper, we propose a new clustering technique whose goal is that of producing both intracluster graphs and intercluster graph with desired topological properties. We formalize this concept in the ▫$(X,Y)$▫-clustering framework, where ▫$Y$▫ is the class that defines the desired topological properties of intracluster graphs and ▫$X$▫ is the class that defines the desired topological properties of the intercluster graph. By exploiting this approach, hybrid visualization tools can effectively combine different node-link and matrix-based representations, allowing users to interactively explore the graph by expansion/contraction of clusters without loosing their mental map. As a proof of concept, we describe the system Visual Hybrid ▫$(X,Y)$▫-clustering (VHYXY) that implements our approach and we present the results of case studies to the visual analysis of social networks.
COBISS.SI-ID: 16097881