Radius–essential Edges in a graph

The graph resulting from contracting edge "e" is denoted as G/e and the graph resulting from deleting edge "e" is denoted as G-e. An edge "e" is radius-essential if rad(G/e) < rad(G), radiusincreasing if rad(G-e)>rad(G), and radius-vital if it is both radius-essential and radius-increasing. We partition the edges that are not radius-vital into three categories. In this paper, we study realizability questions relating to the number of edges that are not radius-vital in the three defined categories. A graph is radius-vital if all its edges are radius-vital. We give a structural characterization of radius-vital graphs

A Survey on eccentric digraphs of (di)graphs

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 survey we take a look on the progress made till date in the theory of Eccentric digraphs of graphs and digraphs, in general. And list the open problems in the area

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

New Results on Edge Rotation Distance Graphs

The concept of edge rotations and distance between graphs was introduced by Gary Chartrand et al. [3]. A graph G can be transformed into a graph H by an edge rotation if G contains distinct vertices u,v and w such that uv 2 E(G), uw =2 E(G) and H =G􀀀uv +uw. In this case, G is transformed into H by “rotating†the edge uv of G into uw. Let S = G1, G2,...,Gk be a set of graphs all of the same order and the same size. Then the rotation distance graph D(S) of S has S as its vertex set and vertices (graphs) Gi and Gj are adjacent if dr(Gi,Gj ) =1, where dr(Gi, Gj ) is the rotation distance between Gi and Gj .A graph G is a edge rotation distance graph(ERDG) (or r - distancegraph) if G =D(S) for some set S of graphs. In [8]Huilgol et al. have showed that the Generalized Petersen Graph Gp(n; 1), the generalized star Km(1; n) is a ERDG. In this paper we consider rotations on some particular graphs Triangular Snake, Double Triangular Snake, Alternating Double Triangular Snake, Quadrilateral Snake, Double Quadrilateral Snake, Alternating Double Quadrilateral snake followed by some generalresults

Cyclic edge extensions-self centered graphs

The eccentricity e(u) of a vertex u is the maximum distance of u to any other vertex of G. The maximum and the minimum eccentricity among the vertices of a graph G are known as the diameter and the radius of G respectively. If they are equal then the graph is said to be a self - centered graph. Edge addition /extension to a graph either retains or changes the parameter of a graph, under consideration. In this paper mainly, we consider edge extension for cycles, with respect to the self-centeredness(of cycles),that is, after an edge set is added to a self centered graph the resultant graph is also a self-centered graph. Also, we have other structural results for graphs with edge -extensions

Square Sum Labeling of Disjoint Union of Graphs

In this paper we prove that if G1 and G2 are square sum, then G1 union G2 union G3 is also square sum, where G3 is a set of isolated vertices

On Some Edge Rotation Distance Graphs

The concept of edge rotations and distance between graphs was introduced by Gary Chartrand et.al [1].A graph G can be transformed into a graph H by an edge rotation if G contains distinct vertices u, v and w such uvE(G) and uwE(G) and H G uv uw . In this case, G is transformed into H by” rotating” the edge uv of G into uw. In this paper we consider rotations on generalized Petersen graphs and minimum selfcenteredgraphs. We have also developed algorithms to generate distance degree injective (DDI) graphs and almost distance degree injective (ADDI) graphs from cycles using the concept of rotations followed by some general results

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

Edge Jump Distance Graphs

The concept of edge jump between graphs and distance between graphs was introduced by Gary Chartrand et al. in [5]. A graph H is obtained from a graph G by an edge jump if G contains four distinct vertices u, v, w, and x such that uv belongs to&nbsp; E(G), wx does not belong to E(G) and H isomorphic to G ¢â‚¬uv + wx. The concept of edge rotations and distance between graphs was first introduced by Chartrand et.al [4]. A graph H is said to be obtained from a graph G by a single edge rotation if G contains three distinct vertices u, v, and w such that uv belongs to \ ‚&nbsp;E(G) and uw does not belong to ‚&nbsp;E(G). If a graph H is obtained from a graph G by a sequence of edge jumps, then G is said to be j-transformed into H.&nbsp;In this paper we consider edge jumps on generalized Petersen graphs Gp(n,1) and cycles. We have also developed an algorithm that gives self-centered graphs and almost self-centered graphs through edge jumps followed by some general results on edge jum &nbsp

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