Next Article in Journal
RIM4J: An Architecture for Language-Supported Runtime Measurement against Malicious Bytecode in Cloud Computing
Previous Article in Journal
Complex Fuzzy Geometric Aggregation Operators
Article Menu
Issue 7 (July) cover image

Export Article

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

On the Diameter and Incidence Energy of Iterated Total Graphs

1
Grupo de Investigación Deartica, Universidad del Sinú, Elías Bechara Zainúm, Cartagena 130001, Colombia
2
Departamento de Matemática, Universidad de Tarapacá, Arica 1000000, Chile
3
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
Full-Text   |   PDF [273 KB, uploaded 2 July 2018]   |  

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 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
Figures

Figure 1

This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited (CC BY 4.0).
SciFeed

Share & Cite This Article

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.

Show more citation formats Show less citations formats

Note that from the first issue of 2016, MDPI journals use article numbers instead of page numbers. See further details here.

Related Articles

Article Metrics

Article Access Statistics

1

Comments

[Return to top]
Symmetry EISSN 2073-8994 Published by MDPI AG, Basel, Switzerland RSS E-Mail Table of Contents Alert
Back to Top