{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T15:44:52Z","timestamp":1760197492517,"version":"build-2065373602"},"reference-count":26,"publisher":"MDPI AG","issue":"7","license":[{"start":{"date-parts":[[2018,7,2]],"date-time":"2018-07-02T00:00:00Z","timestamp":1530489600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>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\u22652, the k-th iterated total graph of G, Tk(G), is defined recursively as Tk(G)=T(Tk\u22121(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))&gt;k\u2212r,r\u22650. 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.<\/jats:p>","DOI":"10.3390\/sym10070252","type":"journal-article","created":{"date-parts":[[2018,7,2]],"date-time":"2018-07-02T10:56:52Z","timestamp":1530529012000},"page":"252","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["On the Diameter and Incidence Energy of Iterated Total Graphs"],"prefix":"10.3390","volume":"10","author":[{"given":"Eber","family":"Lenes","sequence":"first","affiliation":[{"name":"Grupo de Investigaci\u00f3n Deartica, Universidad del Sin\u00fa, El\u00edas Bechara Zain\u00fam, Cartagena 130001, Colombia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Exequiel","family":"Mallea-Zepeda","sequence":"additional","affiliation":[{"name":"Departamento de Matem\u00e1tica, Universidad de Tarapac\u00e1, Arica 1000000, Chile"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Mar\u00eda","family":"Robbiano","sequence":"additional","affiliation":[{"name":"Departamento de Matem\u00e1ticas, Universidad Cat\u00f3lica del Norte, Antofagasta 1240000, Chile"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Jonnathan","family":"Rodr\u00edguez","sequence":"additional","affiliation":[{"name":"Departamento de Matem\u00e1ticas, Universidad Cat\u00f3lica del Norte, Antofagasta 1240000, Chile"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2018,7,2]]},"reference":[{"key":"ref_1","first-page":"153","article-title":"Mean distance in line graph","volume":"32","author":"Buckley","year":"1981","journal-title":"Congr. Numer."},{"key":"ref_2","first-page":"683","article-title":"On Wiener index of graphs and their line graph","volume":"64","author":"Cohen","year":"2010","journal-title":"MATCH Comm. Math. Comput. Chem."},{"key":"ref_3","first-page":"14","article-title":"More on distance of line graph","volume":"33","author":"Gutman","year":"1997","journal-title":"Graph Theory Notes N. Y."},{"key":"ref_4","first-page":"49","article-title":"Distance of line graph","volume":"31","author":"Gutman","year":"1996","journal-title":"Graph Theory Notes N. Y."},{"key":"ref_5","first-page":"105","article-title":"On diameter of line graphs","volume":"8","author":"Ramane","year":"2013","journal-title":"Iran. J. Math. Sci. Inform."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"541","DOI":"10.1021\/ci950143i","article-title":"Topological indices based on the line graph of the molecular graph","volume":"36","author":"Gutman","year":"1996","journal-title":"J. Chem. Inf. Comput. Sci."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"577","DOI":"10.2298\/JSC0008577G","article-title":"On the application of line graphs in quantitative structure-property studies","volume":"65","author":"Gutman","year":"2000","journal-title":"J. Serb. Chem. Soc."},{"key":"ref_8","unstructured":"Gutman, I., and Furtula, B. (2010). Edge versions of topological indices. Novel Molecular Structure Descriptor-Theory and Applications II, University of Kragujevac."},{"key":"ref_9","first-page":"663","article-title":"The edge versions of the Weiner index","volume":"61","author":"Iranmanesh","year":"2009","journal-title":"MATCH Comm. Math. Comput. Chem."},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Dassios, I., O\u2019Keeffe, G., and Jivkov, A.P. (2018). A mathematical model for elasticity using calculus on discrete manifolds. Math. Meth. Appl. Sci., 1\u201314.","DOI":"10.1002\/mma.4892"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"27","DOI":"10.1016\/j.cam.2015.08.017","article-title":"A mathematical model for plasticity and damage: A discrete calculus formulation","volume":"312","author":"Dassios","year":"2017","journal-title":"J. Comput. Appl. Math."},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Dassios, I.K. (2018). Stability of Bounded Dynamical Networks with Symmetry. Symmetry, 10.","DOI":"10.3390\/sym10040121"},{"key":"ref_13","first-page":"573","article-title":"On incidence energy of graphs","volume":"62","author":"Gutman","year":"2009","journal-title":"MATCH Commun. Math. Comput. Chem."},{"key":"ref_14","first-page":"561","article-title":"Incidence energy of a graph","volume":"62","author":"Jooyandeh","year":"2009","journal-title":"MATCH Comm. Math. Comput. Chem."},{"key":"ref_15","first-page":"11","article-title":"Eigenvalue Bounds for the Signless Laplacian","volume":"95","author":"Rowlinson","year":"2007","journal-title":"Publications de l\u2019Institut Math\u00e9matique"},{"key":"ref_16","first-page":"59","article-title":"On a conjeture of the diameter of line graph of graph of diameter two","volume":"36","author":"Ramane","year":"2012","journal-title":"Kragujev. J. Math."},{"key":"ref_17","first-page":"390","article-title":"Iterated line graphs","volume":"33","author":"Buckley","year":"1981","journal-title":"Congr. Numer."},{"key":"ref_18","first-page":"33","article-title":"The size of iterated line graphs","volume":"25","author":"Buckley","year":"1993","journal-title":"Graph Theory Notes N. Y."},{"key":"ref_19","first-page":"49","article-title":"Spectrum of the total graph of a graph","volume":"16","year":"1973","journal-title":"Publ. Inst. Math."},{"key":"ref_20","unstructured":"Cvetkovi\u0107, D.N., Doob, M., and Sachs, H. (1979). Spectra of Graphs: Theory and Applications, Deutscher Verlag der Wissenschaften."},{"key":"ref_21","unstructured":"Cvetkovi\u0107, D.M., Doob, M., Gutman, I., and Torga\u015bev, A. (1988). Recent Results in the Theory of Graph Spectra, Elsevier."},{"key":"ref_22","unstructured":"Godsil, C.D. (1993). Algebraic Combinatorics, Chapman and Hall."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"205","DOI":"10.1023\/A:1008352030960","article-title":"Large families of cospectral graphs","volume":"21","author":"Seress","year":"2000","journal-title":"Des. Codes Cryptogr."},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Brouwer, A.E., Cohen, A.M., and Neumaier, A. (1989). Distance-Regular Graphs, Springer.","DOI":"10.1007\/978-3-642-74341-2"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"91","DOI":"10.1080\/03081089508818382","article-title":"Graphs cospectral with distance-regular graphs","volume":"39","author":"Haemers","year":"1995","journal-title":"Linear Multilinear Algebra"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"189","DOI":"10.1023\/A:1013847004932","article-title":"Spectral characterizations of some distance-regular graphs","volume":"15","author":"Haemers","year":"2002","journal-title":"J. Algebraic Comb."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/10\/7\/252\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T15:10:58Z","timestamp":1760195458000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/10\/7\/252"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2018,7,2]]},"references-count":26,"journal-issue":{"issue":"7","published-online":{"date-parts":[[2018,7]]}},"alternative-id":["sym10070252"],"URL":"https:\/\/doi.org\/10.3390\/sym10070252","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2018,7,2]]}}}