Planarity of Eccentric Digraphs of graphs

The eccentricity e(u) of a vertex u is the maximum distance of u to any othervertex 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 thedigraph that has the same vertex set 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 planarity of eccentric digraph of a graph

On Edge-Distance and Edge-Eccentric graph of a graph

An elementary circuit (or tie) is a subgraph of a graph and the set of edges in this subgraphis called an elementary tieset. The distance d(ei, ej ) between two edges in an undirected graph is defined as the minimum number of edges in a tieset containing ei and ej . The eccentricity ετ (ei) of an edge ei is ετ (ei) = maxej∈Ed(ei, ej ). In this paper, we have introduced the edge - self centered and edge - eccentric graph of a graph and have obtained results on these concepts

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

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

Eccentric Coloring in graphs

he \emph{eccentricity} $e(u)$ of a vertex $u$ is the maximum distance of $u$ to any other vertex of $G$. A vertex $v$ is an \emph{eccentric vertex} of vertex $u$ if the distance from $u$ to $v$ is equal to $e(u)$. An \emph{eccentric coloring} of a graph $G = (V, E)$ is a function \emph{color}: $V \rightarrow N$ such that\\ (i) for all $u, v \in V$, $(color(u) = color(v)) \Rightarrow d(u, v) > color(u)$.\\ (ii) for all $v \in V$, $color(v) \leq e(v)$.\\ The \emph{eccentric chromatic number} $\chi_{e}\in N$ for a graph $G$ is the lowest number of colors for which it is possible to eccentrically color \ $G$ \ by colors: $V \rightarrow \{1, 2, \ldots , \chi_{e} \}$. In this paper, we have considered eccentric colorability of a graph in relation to other properties. Also, we have considered the eccentric colorability of lexicographic product of some special class of graphs

Embedding in Distance Degree Regular and Distance Degree Injective graphs

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 u 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 u 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 graph is distance degree injective (DDI) graph if no two vertices have the same dds. In this paper, we consider the construction of a DDR graph having any given graph G as its induced subgraph. Also we consider construction of some special class of DDI graphs. Keywords: Distance degree sequence, Distance degree regular (DDR) graphs, Almost DDR graphs, Distance degree injective(DDI) grap

Products of distance degree regular and distance degree injective graphs.

The eccentricity e (u) of a vertex u is the maximum distance of u to any other vertex in 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 u in G. Thus the sequence is the dds of the vertex vi in G where 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 graph is distance degree injective (DDI) graph if no two vertices have same dds. In this paper we consider Cartesian and normal products of DDR and DDI graphs. Some structural results have been obtained along with some characterizations

On Marker Set Distance Laplacian Eigenvalues in Graphs

In our previous paper, we had introduced the marker set distance matrix and its eigenvalues. In this paper, we extend them naturally to the Laplacian eigenvalues. To define the Laplacian, we have defined the distance degree sequence of the marker set in the graph. Here we have considered the study of the Laplacian matrix, its characteristic polynomial and related results

Properties of Characteristic Polynomial of Marker Set Distance and its Laplacian

In our previous papers, we had introduced the marker set distance matrix and its eigenvalues and the marker set Laplacian eigenvalues. Also, expressions for the characteristic polynomials of the marker set distance matrix and its Laplacian had been found. In this paper, we discuss the properties of the characteristic polynomials of M-set distance matrix and its Laplacia

Near-field optical power transmission of dipole nano-antennas

Nano-antennas in functional plasmonic applications require high near-field optical power transmission. In this study, a model is developed to compute the near-field optical power transmission in the vicinity of a nano-antenna. To increase the near-field optical power transmission from a nano-antenna, a tightly focused beam of light is utilized to illuminate a metallic nano-antenna. The modeling and simulation of these structures is performed using 3-D finite element method based full-wave solutions of Maxwell’s equations. Using the optical power transmission model, the interaction of a focused beam of light with plasmonic nanoantennas is investigated. In addition, the tightly focused beam of light is passed through a band-pass filter to identify the effect of various regions of the angular spectrum to the near-field radiation of a dipole nano-antenna. An extensive parametric study is performed to quantify the effects of various parameters on the transmission efficiency of dipole nano-antennas, including length, thickness, width, and the composition of the antenna, as well as the wavelength and half-beam angle of incident light. An optimal dipole nanoantenna geometry is identified based on the parameter studies in this work. In addition, the results of this study show the interaction of the optimized dipole nano-antenna with a magnetic recording medium when it is illuminated with a focused beam of light