Almost self-centered graphs

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

doi:10.1007/s10114-011-9628-3

Cited by 24 publications

(17 citation statements)

References 12 publications

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“…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: 95%

“…An almost self-centered graph (ASC graph for short) [20] is a graph in which all but two vertices have the eccentricity equal to r(G). Clearly, d(G) = r(G) + 1 for any ASC graph G. The first Zagreb index M 1 of graph G (see the recent papers [7,29] and the references therein) is among the oldest and the most famous topological index and is defined as…”

Section: Some Relations Between Cei and Eci Of Graphsmentioning

confidence: 99%

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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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Section: Introductionmentioning

confidence: 95%

Section: Some Relations Between Cei and Eci Of Graphsmentioning

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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“…It was proved in [13, Corollary 4.1] that θ r (G) ≤ 2r + 1 holds for any graph G. In this section we first sharpen this result as follows. Interestingly, the below construction which gives a better upper bound is simpler than the construction from [13].…”

Section: Improving Upper Bounds On the R-asc Indexmentioning

confidence: 99%

“…Among other results it was proved in [13] that for any connected graph G and any r ≥ 2 there exists an r-ASC graph which contains G as an induced subgraph. Consequently, the r-ASC index θ r (G) of G was introduced as the minimum number of vertices needed to add to G in order to obtain an r-ASC graph that contains G as an induced subgraph.…”

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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“…In some situations, however, we would like to have certain resources not to lie in the center of a graph. With this motivation, almost self-centered graphs were introduced in [17] as the graphs with exactly two non-central vertices. In the seminal paper constructions that produce almost self-centered graphs are described, and embeddings of graphs into smallest almost self-centered graphs are considered.…”

Section: Introductionmentioning

confidence: 99%

Almost Self-Centered Median and Chordal Graphs

Balakrishnan¹,

Brešar²,

Changat³

et al. 2012

Taiwanese J. Math.

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Abstract. Almost self-centered graphs were recently introduced as the graphs with exactly two non-central vertices. In this paper we characterize almost selfcentered graphs among median graphs and among chordal graphs. In the first case P 4 and the graphs obtained from hypercubes by attaching to them a single leaf are the only such graphs. Among chordal graph the variety of almost self-centered graph is much richer, despite the fact that their diameter is at most 3. We also discuss almost self-centered graphs among partial cubes and among k-chordal graphs, classes of graphs that generalize median and chordal graphs, respectively. Characterizations of almost self-centered graphs among these two classes seem elusive.

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Almost self-centered graphs

Cited by 24 publications

References 12 publications

Some extremal results on the connective eccentricity index of graphs

Some extremal results on the connective eccentricity index of graphs

Embeddings into almost self-centered graphs of given radius

Almost Self-Centered Median and Chordal Graphs

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