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Symmetry 2018, 10(7), 252;

On the Diameter and Incidence Energy of Iterated Total Graphs

Grupo de Investigación Deartica, Universidad del Sinú, Elías Bechara Zainúm, Cartagena 130001, Colombia
Departamento de Matemática, Universidad de Tarapacá, Arica 1000000, Chile
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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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 k2, the k-th iterated total graph of G, Tk(G), is defined recursively as Tk(G)=T(Tk1(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))>kr,r0. 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
Keywords: total graph; line graph; diameter; incidence energy; regular graph total graph; line graph; diameter; incidence energy; regular graph

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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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