Distance Degree Regular Graphs and Theireccentric Digraphs

The eccentricity e(u) of a vertex u is the maximum distance of u to any other vertex of G.The distance degree sequence (dds) of a vertex v in a graph G = (V,E) is a list of the number of vertices at distance 1, 2, . . . , e(u) in that order, where e(u) denotes the eccentricity of v in G. Thus the sequence (di0 , di1 , di2 , . . . , dij , . . .) is the dds of the vertex vi in G where dij denotes number of vertices at distance j from vi. A graph is distance degree regular (DDR) graph if all vertices have the same dds. A vertex v is an eccentric vertex of vertex u if the distance from u to v is equal to e(u). The eccentric digraph ED(G) of a graph (digraph) G is the digraph that has the same vertex as G and an arc from u to v exists in ED(G) if and only if v is an eccentric vertex of u in G. In this paper, we consider the construction of new families of DDR graphs with arbitrary diameter. Also we consider some special class of DDR graphs in relation with eccentric digraph of a graph. Different structural properties of eccentric digraphs of DDR graphs are dealt herewith

Distance Degree Regular Graphs and Distance Degree Injective Graphs: An Overview

The distance d ( v , u ) from a vertex v of G to a vertex u is the length of shortest v to u path. The eccentricity e v of v is the distance to a farthest vertex from v . If d ( v , u ) = e ( v ) , ( u ≠ v ) , we say that u is an eccentric vertex of v . The radius rad ( G ) is the minimum eccentricity of the vertices, whereas the diameter diam ( G ) is the maximum eccentricity. A vertex v is a central vertex if e ( v ) = r a d ( G ) , and a vertex is a peripheral vertex if e ( v ) = d i a m ( G ) . A graph is self-centered if every vertex has the same eccentricity; that is, r a d ( G ) = d i a m ( G ) . The distance degree sequence (dds) of a vertex v in a graph G = ( V , E ) is a list of the number of vertices at distance 1 , 2 , . . . . , e ( v ) in that order, where e ( v ) denotes the eccentricity of v in G . Thus, the sequence ( d i 0 , d i 1 , d i 2 , … , d i j , … ) is the distance degree sequence of the vertex v i in G where d i j denotes the number of vertices at distance j from v i . The concept of distance degree regular (DDR) graphs was introduced by Bloom et al., as the graphs for which all vertices have the same distance degree sequence. By definition, a DDR graph must be a regular graph, but a regular graph may not be DDR. A graph is distance degree injective (DDI) graph if no two vertices have the same distance degree sequence. DDI graphs are highly irregular, in comparison with the DDR graphs. In this paper we present an exhaustive review of the two concepts of DDR and DDI graphs. The paper starts with an insight into all distance related sequences and their applications. All the related open problems are listed

Products and Eccentric digraphs

The eccentricity e(u) of a vertex u is the maximum distance of u to any other vertex of G. A vertex v is an eccentric vertex of vertex u if the distance from u to v is equal to e(u). The eccentric digraph ED(G) of a graph(digraph) G is the digraph that has the same vertex as G and an arc from u to v exists in ED(G) if and only if v is an eccentric vertex of u in G. In this paper, we consider the eccentric digraphs of different products of graphs, viz., cartesian, normal, lexicographic, prism, et

Products and Eccentric Diagraphs

The eccentricity e(u) of a vertex u is the maximum distance of u to any other vertex of G. A vertex v is an eccentric vertex of vertex u if the distance from u to v is equal to e(u). The eccentric digraph ED(G) of a graph(digraph) G is the digraph that has the same vertex as G and an arc from u to v exists in ED(G) if and only if v is an eccentric vertex of u in G. In this paper, we consider the eccentric digraphs of different products of graphs, viz., cartesian, normal, lexicographic, prism, etc

Non-circular features in Saturn's D ring: D68

D68 is a narrow ringlet located only 67,627 km (1.12 planetary radii) from Saturn's spin axis. Images of this ringlet obtained by the Cassini spacecraft reveal that this ringlet exhibits persistent longitudinal brightness variations and a substantial eccentricity (ae=25+/-1 km). By comparing observations made at different times, we confirm that the brightness variations revolve around the planet at approximately the local orbital rate (1751.6 degrees/day), and that the ringlet's pericenter precesses at 38.243+/-0.008 degrees/day, consistent with the expected apsidal precession rate at this location due to Saturn's higher-order gravitational harmonics. Surprisingly, we also find that the ringlet's semi-major axis appears to be decreasing with time at a rate of 2.4+/-0.4 km/year between 2005 and 2013. A closer look at these measurements, along with a consideration of earlier Voyager observations of this same ringlet, suggests that the mean radius of D68 moves back and forth, perhaps with a period of around 15 Earth years or about half a Saturn year. These observations could place important constraints on both the ringlet's local dynamical environment and the planet's gravitational field.Comment: 39 Pages, 11 Figures accepted for publication in Icarus Text slightly modified to match corrections to proof