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

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

In the paper, the authors prove the following theorem. Let Γ be a connected Cayley graph of a dihedral group D2n admitting an arc-regular action of a subgroup D2n≤G≤Aut(Γ) such that every cyclic subgroup of index 2 in D2n is core-free in G. Then Γ is isomorphic to the lexicographic product of the tensor product Kn1⊗⋯⊗Knt by Kcm, where 2n=mn1⋯nt with n1,…,nt pairwise coprime. As the authors point out, this theorem gives only a possible structure for such a Cayley graph Γ. The only known examples are K4, K4[Kc2], and Kn,n=K2[Kcn]. Finally, some applications of the main theorem are given.eorem are given.

COBISS.SI-ID: 102427707