Geodesic
More on Signed Graphs with at Most Three Eigenvalues
Discussiones Mathematicae. Graph Theory, Tome 42 (2022) no. 4, pp. 1313-1331.

Voir la notice de l'article provenant de la source Library of Science

We consider signed graphs with just 2 or 3 distinct eigenvalues, in particular (i) those with at least one simple eigenvalue, and (ii) those with vertex-deleted subgraphs which themselves have at most 3 distinct eigenvalues. We also construct new examples using weighing matrices and symmetric 3-class association schemes.
Keywords: adjacency matrix, simple eigenvalue, strongly regular signed graph, vertex-deleted subgraph, weighing matrix, association scheme 
@article{DMGT_2022_42_4_a16,
     author = {Ramezani, Farzaneh and Rowlinson, Peter and Stani\'c, Zoran},
     title = {More on {Signed} {Graphs} with at {Most} {Three} {Eigenvalues}},
     journal = {Discussiones Mathematicae. Graph Theory},
     pages = {1313--1331},
     publisher = {mathdoc},
     volume = {42},
     number = {4},
     year = {2022},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/DMGT_2022_42_4_a16/}
}
TY  - JOUR
AU  - Ramezani, Farzaneh
AU  - Rowlinson, Peter
AU  - Stanić, Zoran
TI  - More on Signed Graphs with at Most Three Eigenvalues
JO  - Discussiones Mathematicae. Graph Theory
PY  - 2022
SP  - 1313
EP  - 1331
VL  - 42
IS  - 4
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/DMGT_2022_42_4_a16/
LA  - en
ID  - DMGT_2022_42_4_a16
ER  - 
%0 Journal Article
%A Ramezani, Farzaneh
%A Rowlinson, Peter
%A Stanić, Zoran
%T More on Signed Graphs with at Most Three Eigenvalues
%J Discussiones Mathematicae. Graph Theory
%D 2022
%P 1313-1331
%V 42
%N 4
%I mathdoc
%U http://geodesic.mathdoc.fr/item/DMGT_2022_42_4_a16/
%G en
%F DMGT_2022_42_4_a16
Ramezani, Farzaneh; Rowlinson, Peter; Stanić, Zoran. More on Signed Graphs with at Most Three Eigenvalues. Discussiones Mathematicae. Graph Theory, Tome 42 (2022) no. 4, pp. 1313-1331. http://geodesic.mathdoc.fr/item/DMGT_2022_42_4_a16/

[1] M. Anđelić, T. Koledin and Z. Stanić, On regular signed graphs with three eigenvalues, Discuss. Math. Graph Theory 40 (2020) 405–416. https://doi.org/10.7151/dmgt.2279

[2] J. Araujo and T. Bratten, On the spectrum of the Johnson graphs J (n, k, r), Congr. “Dr. Antonio A.R. Monteiro” XIII (2015) 57–62.

[3] P.J. Cameron and J.H. van Lint, Designs, Graphs, Codes and their Links (Cambridge University Press, Cambridge, 1991). https://doi.org/10.1017/CBO9780511623714

[4] X.-M. Cheng, A.L. Gavrilyuk, G.R.W. Greaves and J.H. Koolen, Biregular graphs with three eigenvalues, European J. Combin. 56 (2016) 57–80. https://doi.org/10.1016/j.ejc.2016.03.004

[5] X.-M. Cheng, G.R.W. Greaves and J.H. Koolen, Graphs with three eigenvalues and second largest eigenvalue at most 1, J. Combin. Theory Ser. B 129 (2018) 55–78. https://doi.org/10.1016/j.jctb.2017.09.004

[6] D. Cvetković, P. Rowlinson and S. Simić, An Introduction to the Theory of Graph Spectra (Cambridge University Press, Cambridge, 2010).

[7] E.R. van Dam, Nonregular graphs with three eigenvalues, J. Combin. Theory Ser. B 73 (1998) 101–118. https://doi.org/10.1006/jctb.1998.1815

[8] T. Koledin and Z. Stanić, On a class of strongly regular signed graphs, Publ. Math. Debrecen 97 (2020) 353–365. https://doi.org/10.5486/PMD.2020.8760

[9] L. Lovász, On the Shannon capacity of a graph, IEEE Trans. Inform. Theory 25 (1979) 1–7. https://doi.org/10.1109/TIT.1979.1055985

[10] T.-T. Lu and S.-H. Shiou, Inverses of 2 × 2 block matrices, Comput. Math. Appl. 43 (2002) 119–129. https://doi.org/10.1016/S0898-1221(01)00278-4

[11] F. Ramezani, Some regular signed graphs with only two distinct eigenvalues, Linear Multilinear Algebra (2020), in press. https://doi.org/10.1080/03081087.2020.1736979

[12] F. Ramezani and Y. Bagheri, Constructing strongly regular signed graphs, Ars. Combin., in press.

[13] F. Ramezani, P. Rowlinson and Z. Stanić, On eigenvalue multiplicity in signed graphs, Discrete Math. 343 (2020) 111982. https://doi.org/10.1016/j.disc.2020.111982

[14] F. Ramezani, P. Rowlinson and Z. Stanić, Signed graphs with at most three eigenvalues, Czechoslovak Math. J., in press.

[15] P. Rowlinson, The main eigenvalues of a graph: A survey, Appl. Anal. Discrete Math. 1 (2007) 445–471. https://doi.org/10.2298/AADM0702445R

[16] P. Rowlinson, On graphs with just three distinct eigenvalues, Linear Algebra Appl. 507 (2016) 462–473. https://doi.org/10.1016/j.laa.2016.06.031

[17] P. Rowlinson, More on graphs with just three distinct eigenvalues, Appl. Anal. Discrete Math. 11 (2017) 74–80. https://doi.org/10.2298/AADM161111033R

[18] Z. Stanić, On strongly regular signed graphs, Discrete Appl. Math. 271 (2019) 184–190. https://doi.org/10.1016/j.dam.2019.06.017

[19] Z. Stanić, Main eigenvalues of real symmetric matrices with application to signed graphs, Czechoslovak Math. J. 70 (2020) 1091–1102. https://doi.org/10.21136/CMJ.2020.0147-19

[20] T. Zaslavsky, Matrices in the theory of signed simple graphs, in: Advances in Discrete Mathematics and Applications, Mysore 2008, B.D. Acharya, G.O.H. Katona, J. Nešetřil (Ed(s)), (Mysore, Ramanujan Math. Soc., 2010) 207–229.