Volume 66 | Issue 7 | Year 2020 | Article Id. IJMTT-V66I7P520 | DOI : https://doi.org/10.14445/22315373/IJMTT-V66I7P520

For a finite group G, the power graph P(G) is a simple connected graph having vertex set as the set of elements of finite group G, where two distinct vertices are adjacent if and only if one is a power of the other. In this paper, we obtain the distance spectrum power graphs of finite groups such as cyclic groups, dihedral groups, dicyclic groups, abelian groups, elementary abelian p groups and other non abelian groups.

[1] J. Abawajy, A. Kelarev and M. Chowdhury, Power graphs: A survey, Ele. J. Graph Th. Appl. 1(2) (2013) 125-147.

[2] M. Aouchiche and P. Hansen, Distance spectra of graphs: a survey, Linear Algebra Appl. 458 (2014) 301{386.

[3] S. Banerjee and A. Adhikari, Signless Laplacian spectra of power graphs of certain nite groups, AKCE Int. J. Graphs Comb. DOI:10.1016/j.akcej.2019.03.009 (2019).

[4] A. E. Brouwer, W. H. Haemers Spectra of Graphs, Springer New York 2010.

[5] P. J. Cameron, The power graphs of a finite group II, J. Group Theory 13(6) (2010) 779-783.

[6] P. J. Cameron and S. Ghosh, The power graphs of a finite group, Dicrete Math. 311(13) (2011) 1220-1222.

[7] S. Chattopadhyay and P. Panigrahi, On Laplacian spectrum of power graphs of finite cyclic and dihedral groups, Linear Multi. Algebra 63:7 (2015) 1345-1355.

[8] S. Chattopadhyay and P. Panigrahi, Connectivity and planarity of power graphs of finite cyclic, dihedral and dicyclic groups, Algebra Discrete Math. 18 (2014) 42-49.

[9] I. Chakrabarty, M. Ghosh and M. K. Sen, Undirected power graph of semigroups, Semigroup Forum 78 (2009) 410{426.

[10] T. T. Chelvan and M. Sattanathan, Power graphs of finite abelian groups, Algebra Discrete Math. 16(1) (2013) 33{41.

[11] D. M. Cvetkovic, P. Rowlison and S. Simic, An Introduction to Theory of Graph spectra Spectra of graphs. Theory and application, Lon. Math. S. student Text, 75. Cambridge University Press, Inc. UK 2010. On distance spectra of power graphs of finite groups 11

[12] A. Hamzeh and A. R. Ashrafi, Spectrum ans L-spectrum of the power graph and its main supergraph for certain finite groups Filomat, 31(16) (2017) 5323{5334.

[13] R. Horn and C. Johnson, Matrix Analysis Second Edition, Cambridge University Press 2013.

[14] A. V. Kelarev and S. J. Quinn, Directed graphs and combinatorial properties of semigroups, J.Algebra (251) (2002) 16-26.

[15] Z. Mehranian, A. Gholami and A. R. Ashrafi, A note on the power graph of a finite group, Int. J. of Group Theory 5 (2016) 1{10.

[16] Z. Mehranian, A. Gholami and A. R. Ashrafi, The spectra of power graphs of certain finite groups, Linear Multi. Algebra 65(5) (2016) 1003{1010.

[17] W. K. Nicholson, Introduction to abstract algebra, fourth edition, John Wiley and sons, New Jersey (2012).

[18] R. Rajkumar and T. Anitha, Laplacian spectrum of reduced power graph of certain finite groups, Linear Multi. Algebra DOI:10.1080/03081087.2019.1636930 (2019).

[19] R. P. Panda, Laplacian spectra power graphs of certain finite groups, Graphs and Comb. DOI:10.1007/s00373-019-02070-x (2019).

[20] D. Stevanovic, Large sets of long distance equienergetic graphs, Ars Math. Contemp. 2(1) (2009).

Mudasir A. Wani,A. K. Shrivastav, "On distance spectra of power graphs of finite groups," *International Journal of Mathematics Trends and Technology (IJMTT)*, vol. 66, no. 7, pp. 150-160, 2020. *Crossref*, https://doi.org/10.14445/22315373/IJMTT-V66I7P520