### On the generalized distance spectrum of graphs

LU Peng-li*, LIU Wen-zhi

1. School of Computer and Communication, Lanzhou University of Technology, Lanzhou 730050, Gansu, China
• Published:2020-09-17

Abstract: The upper and lower bounds of the generalized distance spectral radius of G and of its line graph L(G)are obtained, based on some graph parameters, and the extremal graphs are determined. Then, the generalized distance spectrum of some composite graphs is calculated.

CLC Number:

• O157.5
 [1] AOUCHICHE M, HANSEN P. Two Laplacians for the distance matrix of a graph[J]. Linear Algebra and Its Applications, 2013, 430:21-33.[2] TIAN Guixian, CUI Shuyu, HE Jingxiang. The generalized distance matrix[J]. Linear Algebra and Its Applications, 2019, 563:1-23.[3] CVETKOVIC D M, DOOB M, SACHS H. Spectra of graphs-theory and application[M]. New York: Academic Press, 1980.[4] WOODHOUSE J H. On Rayleighs principle[J]. Geophysical Journal International, 2007, 46(1):11-22.[5] AOUCHICHE M, HANSEN P. Distance spectra of graphs: a survey[J]. Linear Algebra and Its Applications, 2014, 458:301-386.[6] AOUCHICHE M, HANSEN P. Some properties of the distance Laplacian eigenvalues of a graph[J]. Czechoslovak Math, 2014, 64:751-761.[7] ALHEVAZ A, BAGHIPUR M, HASHEMI E, On the distance signless Laplacian spectrum of graphs[J]. Bulletin of the Malaysian Mathematical Sciences Society, 2018. https://doi.org/10.1007/s40840-018-0619-8.[8] ROBERTO C D, GERMAIN P, OSCAR R. New results on the Dα-matrix of connected graphs[J]. Linear Algebra and Its Applications, 2019, 577:168-185.[9] LIN Huiqiu, LU Xiwen. Bounds on the distance signless Laplacian spectral radius in terms of clique number[J]. Linear Multilinear Algebra, 2015, 63:1750-1759.[10] NIU Aihong,FAN Dandan,WANG Guoping. On the distance Laplacian spectral radius of bipartite graphs[J]. Discrete Appl Math, 2015, 186:207-213.[11] TIAN Fenglei, WONG Dein, ROU Jianling. Proof for four conjectures about the distance Laplacian and distance signless Laplacian eigenvalues of a graph[J]. Linear Algebra Appl, 2015, 471:10-20.[12] XING Rundan, ZHOU Bo, LI Jianping. On the distance signless Laplacian spectral radius of graphs[J]. Linear Multilinear Algebra, 2014, 62:1377-1387.[13] MADEN A D, DAS K C, CEVIK A S. Sharp upper bounds on the spectral radius of the signless Laplacian matrix of a graph[J]. Appl Math Comput, 2013, 219:5025-5032.[14] LASKAR R. Eigenvalues of the adjacency matrix of cubic lattice graphs[J]. Pacific Journal of Mathematics, 1969, 29:623-629.[15] HARARY F. Graph theory[M]. New Delhi: Narosa Publishing House, 1999.[16] MINC H. Nonnegative matrices[M]. New York: Wiley, 1988.[17] RAMANE H S, GUTMAN I, GANAGI A B. On diameter of line graphs[J]. Iranian Journal of Mathematical Sciences and Informatics, 2013, 8:105-109.[18] RAMANE H S, GUTMAN I, REVANKAR D S, et al. Distance spectra and distance energies of iterated line graphs of regular graphs[J]. Publ Inst Math, 2009, 85:39-46.[19] FOWLER P W, CAPOROSSI G, HANSEN P. Distance matrices, Wiener indices, and related invariants of fullerenes[J]. J Phys Chem A, 2001, 105:6232-6242.[20] INDULAL G. Distance spectrum of graph compositions[J]. ARS Mathematica Contemporanea, 2009, 2:93-100.
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