Eccentric graphs

Chartrand, Gary; Gu, Wenjin; Schultz, Michelle; Winters, Steven J.

doi:10.1002/(sici)1097-0037(199909)34:2<115::aid-net4>3.0.co;2-k

Cited by 12 publications

(5 citation statements)

References 2 publications

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“…The eccentric digraph of a graph or digraph is a distance based mapping, defined on a relation induced by distances in a graph or digraph, that is also represented by a graph. Many distance based relations can be found in literature in the study of antipodal graphs [31], antipodal digraphs [31], eccentric graphs [32,33], and so forth. The eccentric digraph ED( ) of a graph (or digraph) is the digraph that has the same vertex as and an arc from to V exists in ED( ) if and only if V is an eccentric vertex of in .…”

Section: Relationship With Other Propertiesmentioning

confidence: 99%

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

Huilgol

2014

Journal of Discrete Mathematics

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The distanced(v,u)from a vertexvofGto a vertexuis the length of shortestvtoupath. Theeccentricityevofvis the distance to a farthest vertex fromv. Ifd(v,u) = e(v), (u ≠ v), we say thatuis aneccentric vertexofv. Theradiusrad(G)is theminimum eccentricityof the vertices, whereas thediameterdiam(G)is themaximum eccentricity. A vertexvis acentral vertexife(v) = rad(G), and a vertex is aperipheral vertexife(v) = diam(G). A graph isself-centeredif every vertex has the same eccentricity; that is,rad(G) = diam(G). Thedistance degree sequence (dds)of a vertexvin a graphG = (V, E)is a list of the number of vertices at distance1, 2, ... . , e(v)in that order, wheree(v)denotes the eccentricity ofvinG. Thus, the sequence(di0,di1,di2, …, dij,…)is the distance degree sequence of the vertexviinGwheredijdenotes the number of vertices at distancejfromvi. The concept ofdistance degree regular (DDR) graphswas 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 isdistance degree injective (DDI) graphif 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.

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Section: Relationship With Other Propertiesmentioning

confidence: 99%

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

Huilgol

2014

Journal of Discrete Mathematics

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“…Consequently, different classes of graphs and networks in which the center and the periphery have a special structure were introduced. These classes include self-centered graphs (alias eccentric graphs) [1,3,4,15], their generalization to graphs whose center is a k-distance dominating set [5], and almost self-centered graphs [2,8,10]. The latter graphs (as well as almost peripheral graphs) turned out to be extremal graphs for a newly introduced measure of non-self-centrality introduced and studied in [17].…”

Section: Introductionmentioning

confidence: 99%

Simplified constructions of almost peripheral graphs and improved embeddings into them

Klavžar

Narayankar

et al. 2018

Filomat

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The center and the periphery of a graph are the sets of vertices with minimum and maximum eccentricity, respectively. A graph is called almost peripheral (AP) if all its vertices but one lie in the periphery. The rAP index AP r (G) of a graph G is the smallest number of vertices needed to add to G to obtain an rAP graph in which G lies as an induced subgraph. In this paper new, simplified constructions of AP graphs are presented. It is proved that if r ≥ 2 and n ≥ 2, then AP r (K n) ≤ 4r − 3. Moreover, if G is not complete and has at least three vertices, then AP r (G) ≤ 4r − 4. In this way the previously best know bound AP r (G) ≤ 4r − 2 is improved.

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“…In the case when C(G) = V (G) holds, the graph G is called self-centered or eccentric. These graphs were extensively studied by now, see the survey [4] on the early investigations and a selection of more recent papers [5,8,12]. If a graph is not self-centered, then it contains at least two vertices that do not belong to its center.…”

Section: Introductionmentioning

confidence: 99%

Almost-Peripheral Graphs

Klavžar¹,

Narayankar²,

Walikar³

et al. 2014

Taiwanese J. Math.

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The center C(G) and the periphery P (G) of a connected graph G consist of the vertices of minimum and maximum eccentricity, respectively. Almostperipheral (AP) graphs are introduced as graphs G with |P (G)| = |V (G)|−1 (and |C(G)| = 1). AP graph of radius r is called an r-AP graph. Several constructions of AP graph are given, in particular implying that for any r ≥ 1, any graph can be embedded as an induced subgraph into some r-AP graph. A decomposition of AP-graphs that contain cut-vertices is presented. The r-embedding index Φ r (G) of a graph G is introduced as the minimum number of vertices which have to be added to G such that the obtained graph is an r-AP graph. It is proved that Φ 2 (G) ≤ 5 holds for any non-trivial graphs and that equality holds if and only if G is a complete graph.

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Eccentric graphs

Cited by 12 publications

References 2 publications

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

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

Simplified constructions of almost peripheral graphs and improved embeddings into them

Almost-Peripheral Graphs

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