Almost-Peripheral Graphs

Klavžar, Sandi; Narayankar, Kishori P.; Walikar, H. B.; Lokesh, S. B.

doi:10.11650/tjm.18.2014.3267

Cited by 13 publications

(7 citation statements)

References 10 publications

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“…In [9] a question was posed whether there exist r-AP graphs of order n < 4r + 1 for r ≥ 4. A positive answer to this problem was given in [10] by demonstrating that for any r ≥ 1 there exists an r-AP graph of order 4r − 1.…”

Section: New Constructions Of Ap Graphsmentioning

confidence: 99%

“…In this paper we are interested in almost peripheral graphs that were introduced in [9] and in part motivated by location problems in which it is required that most of the resources do not lie in the center. A graph G is called almost-peripheral, AP for short, if all but one of its vertices lie in the periphery, that is, if |P(G)| = |V(G)| − 1 holds.…”

Section: Introductionmentioning

confidence: 99%

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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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Section: New Constructions Of Ap Graphsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Simplified constructions of almost peripheral graphs and improved embeddings into them

Klavžar

Narayankar

et al. 2018

Filomat

View full text Add to dashboard Cite

show abstract

“…Some nice results on other attractive distance-based topological indices of graphs can be found in [14,16,19,31] for eccentric distance sum, [8] for Zagreb eccentricity indices, [23] for adjacent eccentric distance sum index, [10,18,26,27] for degree distance, [11,15] for Gutman index and a recent survey [28] for extremal problems. Furthermore, some interesting properties of eccentricity are reported in [20,21].…”

Section: Introductionmentioning

confidence: 99%

Some extremal results on the connective eccentricity index of graphs

Das

Liu

2016

Journal of Mathematical Analysis and Applications

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Please cite this article in press as: K. Xu et al., Some extremal results on the connective eccentricity index of graphs, J. Math. Anal. Appl. (2015), http://dx. AbstractThe connective eccentricity index (CEI) of a graph G is defined asare the eccentricity and the degree of vertex v i , respectively, in G. In this paper we obtain some lower and upper bounds on the connective eccentricity index for all trees of order n and with matching number β and characterize the corresponding extremal trees. And the maximal graphs of order n and with matching number β and n edges have been determined which maximize the connective eccentricity index. Also the extremal graphs with maximal connective eccentricity index are completely characterized among all connected graphs of order n and with matching number β. Moreover we establish some relations between connective eccentricity index and eccentric connectivity index, as another eccentricity-based invariant, of graphs. AMS classification (2010): 05C07, 05C12, 05C35

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“…Graphs dual to the almost self-centered graphs were studied in [14] and named almostperipheral graphs (AP graphs for short). Very recently, a measure of non-self-centrality was introduced in [22], where ASC graphs and AP graphs, along with a newly defined weakly AP graphs, play a significant role as extremal graphs in studies of this new measure.…”

Section: Introductionmentioning

confidence: 99%

Embeddings into almost self-centered graphs of given radius

Liu

Das

et al. 2018

J Comb Optim

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A graph is almost self-centered (ASC) if all but two of its vertices are central. An almost self-centered graph with radius r is called an r-ASC graph. The r-ASC index θ r (G) of a graph G is the minimum number of vertices needed to be added to G such that an r-ASC graph is obtained that contains G as an induced subgraph. It is proved that θ r (G) ≤ 2r holds for any graph G and any r ≥ 2 which improves the earlier known bound θ r (G) ≤ 2r +1. It is further proved that θ r (G) ≤ 2r − 1 holds if r ≥ 3 and G is of order at least 2. The 3-ASC index of complete graphs is determined. It is proved that θ 3 (G) ∈ {3, 4} if G has diameter 2 and for several classes of graphs of diameter 2 the exact value of the 3-ASC index is obtained. For instance, if a graph G of diameter 2 does not contain a diametrical triple, then θ 3 (G) = 4. The 3-ASC index of paths of order n ≥ 1, cycles of order n ≥ 3, and trees of order n ≥ 10 and diameter n−2 are also determined, respectively, and several open problems proposed.

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Almost-Peripheral Graphs

Cited by 13 publications

References 10 publications

Simplified constructions of almost peripheral graphs and improved embeddings into them

Simplified constructions of almost peripheral graphs and improved embeddings into them

Some extremal results on the connective eccentricity index of graphs

Embeddings into almost self-centered graphs of given radius

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