## On Composition of Signed Graphs

 K. Shahul Hameed Department of Mathematics Government Brennen College Thalassery - 670106, India. K.A. Germina Research Center and PG Department of Mathematics, Mary Matha Arts and Science College, Mananthavady - 670645, India.

## Abstract

A graph whose edges are labeled either as positive or negative is called a signed graph. In this article, we extend the notion of composition of (unsigned) graphs (also called lexicographic product) to signed graphs. We employ Kronecker product of matrices to express the adjacency matrix of this product of two signed graphs and hence find its eigenvalues when the second graph under composition is net-regular. A signed graph is said to be net-regular if every vertex has constant net-degree, namely, the difference of the number of positive and negative edges incident with a vertex. We also characterize balance in signed graph composition and have some results on the Laplacian matrices of this product.

Keywords: signed graph, eigenvalues, graph composition, regular graphs, net-regular signed graphs

2010 Mathematics Subject Classification: 05C22, 05C50, 05C76.

## References

 [1] B.D. Acharya, Spectral criterion for cycle balance in networks, J. Graph Theory 4 (1980) 1--11, doi: 10.1002/jgt.3190040102. [2] F. Barahona, On the computational complexity of Ising spin glass models, J. Phys. (A) 15 (1982) 3241--3253, doi: 10.1088/0305-4470/15/10/028. [3] D. Cartwright and F. Harary, Structural balance: a generalization of Heider?s theory, Psychological Rev. 63 (1956) 277--293, doi: 10.1037/h0046049 . [4] G. Chartrand, H. Gavlas, F. Harary and M. Schultz, On signed degrees in signed graphs, Czechoslovak Math. J. 44 (1994) 677--680. [5] D.M. Cvetkovi'c, M. Doob and H. Sachs, Spectra of Graphs (third ed., Johann Abrosius Barth Verlag, 1995). [6] F. Harary, Graph Theory ( Addison Wesley, Reading, MA, 1972). [7] K.A. Germina and K. Shahul Hameed , On signed paths, signed cycles and their energies, Applied Math. Sci. 70 (2010) 3455--3466, doi: 10.1016/j.laa.2010.10.026. [8] K.A. Germina, K. Shahul Hameed and T. Zaslavsky, On product and line graphs of signed graphs, their eigenvalues and energy, Linear Algebra Appl. 435 (2011) 2432--2450, doi: 10.1007/s10587-007-0079-z. [9] S. Pirzada, T.A. Naikoo and F.A. Dar, Signed degree sets in signed graphs, Czechoslovak Math. J. 57 (2007) 843--848, doi: 10.1215/S0012-7094-59-02667-5. [10] S. Pirzada, T.A. Naikoo and F.A. Dar, Signed degree sets in signed bipartite graphs, AKCE Int. J. Graphs and Combin. 4(3) (2007) 301--312. [11] G. Sabidussi, The composition of graphs, Duke Math. J. 26 (1959) 693--696, doi: 10.1215/S0012-7094-59-02667-5. [12] G. Sabidussi, The lexicographic product of graphs, Duke Math. J. 28 (1961) 573--578, doi: 10.1215/S0012-7094-61-02857-5. [13] T. Zaslavsky, Signed graphs, Discrete Appl. Math. 4 (1982) 47--74, Erratum: Discrete Appl. Math. 5 (1983) 248--248, doi: 10.1016/0166-218X(83)90047-1. [14] T. Zaslavsky, Matrices in the theory of signed simple. [15] T. Zaslavsky, A mathematical bibliography of signed and gain graphs and allied areas, VII Edition, Electronic J. Combin. 8 (1998), Dynamic Surveys: #8. [16] F. Zhang, Matrix Theory: Basic Theory and Techniques ( Springer-Verlag, 1999).