Document Type : Original Manuscript


Department of Pure Mathematics, Faculty of Mathematical Sciences, University of Kashan, P.O. Box 87317–53153, Kashan, I. R. Iran.


The commuting graph of a finite group $G$, $\mathcal{C}(G)$, is a simple graph with vertex set $G$ in which two vertices $x$ and $y$ are adjacent if and only if $xy = yx$. The aim of this paper is to compute the distance Laplacian spectrum and the distance Laplacian energy of the commuting graph of $CA$-groups.


  1. F. Ali, M. Salman and S. Huang, On the commuting graph of dihedral group, Comm. Algebra, 44 (6) (2016), 2389–2401.
  2. 2. M. Aouchiche and P. Hansen, Two Laplacians for the distance matrix of a graph, Linear Algebra Appl., 439 (2013), 21–3
  3.  M. Aouchiche and P. Hansen, Distance spectra of graphs: A survey, Linear Algebra Appl., 458 (2014), 301–386.
  4.  M. Aouchiche and P. Hansen, Some properties of the distance Laplacian eigenvalues of a graph, Czech. Math. J., 64 (139) (2014), 751–761.
  5.  M. Aouchiche and P. Hansen, Distance Laplacian eigenvalues and chromatic number in graphs, Filomat, 31 (9) (2017), 2545–2555.
  6.  A. R. Ashrafi and M. Torktaz, On the commuting graph of CA-groups, submitted.
  7.  N. Biggs, Algebraic Graph Theory, Second edition, Cambridge University Press, Cambridge, 1993.
  8.  A. E. Brouwer and W. H. Haemers, Eigenvalues and perfect matchings, Linear Algebra Appl., 395 (2005), 155–162.
  9.  H. Lin and B. Zhou, On the distance Laplacian spectral radius of graphs, Linear Algebra Appl., 475 (2015), 265–275.
  10.  M. Mirzargar, P. P. Pach and A. R. Ashrafi, The automorphism group of commuting graph of a finite group, Bull. Korean Math. Soc., 51 (4) (2014), 1145–1153.
  11.  M. Mirzargar and A. R. Ashrafi, Some distance-based topological indices of a non-commuting graph, Hacet. J. Math. Stat., 41 (6) (2012), 515–526.
  12.  M. Torktaz and A. R. Ashrafi, Spectral properties of the commuting graphs of certain groups, AKCE Int. J. Graphs Combin., 16 (2019), 300–309.
  13.  The GAP Team, Gap – Groups, Algorithms, and Programming, version 4.7.5, 2014.
  14.  J. Yang, L. You and I. Gutman, Bounds on the distance Laplacian energy of graphs, Kragujevac J. Math., 37 (2) (2013), 245–255.