Geodesic
New Classes of Simultaneous Cospectral Graphs for Adjacency, Laplacian and Normalized Laplacian Matrices
Kragujevac Journal of Mathematics, Tome 43 (2019) no. 2, p. 303 .

Voir la notice de l'article provenant de la source eLibrary of Mathematical Institute of the Serbian Academy of Sciences and Arts

Butler \cite{But1} constructed simultaneous cospectral graphs for the adjacency and normalized Laplacian matrices, and asked the same for all three matrices, namely, adjacency, Laplacian and normalized Laplacian. In this paper, we determine the full adjacency, Laplacian and normalized Laplacian spectrum of the $Q$-vertex join and $Q$-edge join of a connected regular graph with an arbitrary regular graph in terms of their respective eigenvalues. Applying these results we construct some non-regular $A$-cospectral, $L$-cospectral and $\mathcal{L}$-cospectral graphs which gives a partial answer of the question asked by Butler \cite{But1}. Moreover, we determine the number of spanning trees and the Kirchhoff index of the newly constructed graphs.
Classification : 05C50, 47A75
Keywords: spectrum, cospectral graphs, $Q$-vertex join, $Q$-edge join, spanning tree, Kirchhoff index 
@article{KJM_2019_43_2_a10,
     author = {A. Das and P. Panigrahi},
     title = {New {Classes} of {Simultaneous} {Cospectral} {Graphs} for {Adjacency,} {Laplacian} and {Normalized} {Laplacian} {Matrices}},
     journal = {Kragujevac Journal of Mathematics},
     pages = {303 },
     publisher = {mathdoc},
     volume = {43},
     number = {2},
     year = {2019},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/KJM_2019_43_2_a10/}
}
TY  - JOUR
AU  - A. Das
AU  - P. Panigrahi
TI  - New Classes of Simultaneous Cospectral Graphs for Adjacency, Laplacian and Normalized Laplacian Matrices
JO  - Kragujevac Journal of Mathematics
PY  - 2019
SP  - 303 
VL  - 43
IS  - 2
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/KJM_2019_43_2_a10/
LA  - en
ID  - KJM_2019_43_2_a10
ER  - 
%0 Journal Article
%A A. Das
%A P. Panigrahi
%T New Classes of Simultaneous Cospectral Graphs for Adjacency, Laplacian and Normalized Laplacian Matrices
%J Kragujevac Journal of Mathematics
%D 2019
%P 303 
%V 43
%N 2
%I mathdoc
%U http://geodesic.mathdoc.fr/item/KJM_2019_43_2_a10/
%G en
%F KJM_2019_43_2_a10
A. Das; P. Panigrahi. New Classes of Simultaneous Cospectral Graphs for Adjacency, Laplacian and Normalized Laplacian Matrices. Kragujevac Journal of Mathematics, Tome 43 (2019) no. 2, p. 303 . http://geodesic.mathdoc.fr/item/KJM_2019_43_2_a10/