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Symmetry 2018, 10(7), 252; https://doi.org/10.3390/sym10070252

# On the Diameter and Incidence Energy of Iterated Total Graphs

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Grupo de Investigación Deartica, Universidad del Sinú, Elías Bechara Zainúm, Cartagena 130001, Colombia
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Departamento de Matemática, Universidad de Tarapacá, Arica 1000000, Chile
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Departamento de Matemáticas, Universidad Católica del Norte, Antofagasta 1240000, Chile
*
Author to whom correspondence should be addressed.
Received: 11 June 2018 / Revised: 28 June 2018 / Accepted: 29 June 2018 / Published: 2 July 2018
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# Abstract

The total graph of G, $T(G)$ is the graph whose vertex set is the union of the sets of vertices and edges of G, where two vertices are adjacent if and only if they stand for either incident or adjacent elements in G. For $k≥2$, the k-th iterated total graph of G, $Tk(G)$, is defined recursively as $Tk(G)=T(Tk−1(G)),$ where $T1(G)=T(G)$ and $T0(G)=G.$ If G is a connected graph, its diameter is the maximum distance between any pair of vertices in G. The incidence energy $IE(G)$ of G is the sum of the singular values of the incidence matrix of G. In this paper, for a given integer k we establish a necessary and sufficient condition under which $diam(Tr+1(G))>k−r,$$r≥0$. In addition, bounds for the incidence energy of the iterated graph $Tr+1(G)$ are obtained, provided G is a regular graph. Finally, new families of non-isomorphic cospectral graphs are exhibited. View Full-Text
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MDPI and ACS Style

Lenes, E.; Mallea-Zepeda, E.; Robbiano, M.; Rodríguez, J. On the Diameter and Incidence Energy of Iterated Total Graphs. Symmetry 2018, 10, 252.

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