# zbMATH — the first resource for mathematics

On the sum of $$k$$ largest singular values of graphs and matrices. (English) Zbl 1222.05172
Summary: In the recent years, the trace norm of graphs has been extensively studied under the name of graph energy. The trace norm is just one of the Ky Fan $$k$$-norms, given by the sum of the $$k$$ largest singular values, which are studied more generally in the present paper. Several relations to chromatic number, spectral radius, spread, and to other fundamental parameters are outlined. Some results are extended to more general matrices.

##### MSC:
 05C50 Graphs and linear algebra (matrices, eigenvalues, etc.)
##### Keywords:
Ky fan norms; graph energy; singular values; Hadamard matrices
Full Text:
##### References:
 [1] Bollobás, B., Modern graph theory, Graduate texts in mathematics, Vol. 184, (1998), Springer Verlag, New York, xiv+394 pp. · Zbl 0902.05016 [2] Caporossi, G.; Cvetković, D.; Gutman, I.; Hansen, P., Variable neighborhood search for extremal graphs. 2. finding graphs with extremal energy, J. chem. inf. comput. sci., 39, 984-996, (1999) [3] Ebrahimi, J.; Mohar, B.; Nikiforov, V.; Ahmady, A.S., On the sum of two largest eigenvalues of a symmetric matrix, Linear algebra appl., 429, 2781-2787, (2008) · Zbl 1148.05047 [4] Gregory, D.; Hershkowitz, D.; Kirkland, S., The spread of the spectrum of a graph, Linear algebra appl., 332, 23-35, (2001) · Zbl 0978.05049 [5] Gutman, I., The energy of a graph, Ber. math.-stat. sekt. forschungszent. Graz, 103, 1-22, (1978) [6] Haemers, W.; Xiang, Q., Strongly regular graphs with parameters $$(4 m^4$$, $$2 m^4 + m^2$$, $$m^4 + m^2$$, $$m^4 + m^2)$$ exist for all $$m > 1$$, European J. combin., 31, 1553-1559, (2010) · Zbl 1225.05252 [7] Hoffman, A.J., On eigenvalues and colorings of graphs, graph theory and its applications, (1970), Academic Press New York, pp. 79-91 [8] Horn, R.; Johnson, C., Matrix analysis, (1985), Cambridge University Press Cambridge, xiii+561 pp. · Zbl 0576.15001 [9] Koolen, J.H.; Moulton, V., Maximal energy graphs, Adv. appl. math., 26, 47-52, (2001) · Zbl 0976.05040 [10] Mohar, B., On the sum of k largest eigenvalues of graphs and symmetric matrices, J. combin. theory ser. B, 99, 306-313, (2009) · Zbl 1217.05151 [11] Nikiforov, V., Some inequalities for the largest eigenvalue of a graph, Combin. probab. comput., 11, 179-189, (2002) · Zbl 1005.05029 [12] Nikiforov, V., Linear combinations of graph eigenvalues, Electron. J. linear algebra, 15, 329-336, (2006) · Zbl 1142.05343 [13] E. Nosal, Eigenvalues of Graphs, Master’s thesis, University of Calgary, 1970.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.