# Almost Self-centered Index of Some Graphs

• Priyanka Singh
• Pratima Panigrahi
Conference paper
Part of the Advances in Intelligent Systems and Computing book series (AISC, volume 1097)

## Abstract

For a simple connected graph G, center C(G) and periphery P(G) are subgraphs induced on vertices of G with minimum and maximum eccentricity, respectively. An n-vertex graph G is said to be an almost self-centered (ASC) graph if it contains $$n-2$$ central vertices and two peripheral (diametral) vertices. An ASC graph with radius r is known as an r-ASC graph. The r-ASC index of any graph G is defined as the minimum number of new vertices, and required edges, to be introduced to G such that the resulting graph is r-ASC graph in which G is induced. For $$r=2,3$$, r-ASC index of few graphs is calculated by Klavžar et al. (Acta Mathematica Sinica, 27:2343–2350, 2011 [1]), Xu et al. (J Comb Optim 36(4):1388–1410, 2017 [2]), respectively. Here we give bounds to r-ASC index of diameter two graphs and determine the exact value of this index for paths and cycles.

## Keywords

Radius Diameter Almost self-centered graphs r-ASC embedding index

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© Springer Nature Singapore Pte Ltd. 2020

## Authors and Affiliations

• Priyanka Singh
• 1
• Pratima Panigrahi
• 1
1. 1.Indian Institute of Technology KharagpurKharagpurIndia