Browse > Article
http://dx.doi.org/10.4134/BKMS.b210416

ON EIGENSHARPNESS AND ALMOST EIGENSHARPNESS OF LEXICOGRAPHIC PRODUCTS OF SOME GRAPHS  

Abbasi, Ahmad (Department of Pure Mathematics Faculty of Mathematical Sciences University of Guilan, Center of Excellence for Mathematical Modeling Optimization and Combinatorial Computing (MMOCC) University of Guilan)
Taleshani, Mona Gholamnia (Department of Pure Mathematics Faculty of Mathematical Sciences University of Guilan)
Publication Information
Bulletin of the Korean Mathematical Society / v.59, no.3, 2022 , pp. 685-695 More about this Journal
Abstract
The minimum number of complete bipartite subgraphs needed to partition the edges of a graph G is denoted by b(G). A known lower bound on b(G) states that b(G) ≥ max{p(G), q(G)}, where p(G) and q(G) are the numbers of positive and negative eigenvalues of the adjacency matrix of G, respectively. When equality is attained, G is said to be eigensharp and when b(G) = max{p(G), q(G)} + 1, G is called an almost eigensharp graph. In this paper, we investigate the eigensharpness and almost eigensharpness of lexicographic products of some graphs.
Keywords
Lexicographic product of graphs; biclique partition number; eigensharp graphs;
Citations & Related Records
연도 인용수 순위
  • Reference
1 N. Abreu, D. M. Cardoso, P. Carvalho, and C. T. M. Vinagre, Spectra and Laplacian spectra of arbitrary powers of lexicographic products of graphs, Discrete Math. 340 (2017), no. 1, 3235-3244. https://doi.org/10.1016/j.disc.2016.07.017   DOI
2 R. L. Graham and H. O. Pollak, On the addressing problem for loop switching, Bell System Tech. J. 50 (1971), 2495-2519. https://doi.org/10.1002/j.1538-7305.1971.tb02618.x   DOI
3 F. Harary, On the group of the composition of two graphs, Duke Math. J. 26 (1959), 29-34. http://projecteuclid.org/euclid.dmj/1077468336   DOI
4 G. Sabidussi, The composition of graphs, Duke Math. J. 26 (1959), 693-696. http://projecteuclid.org/euclid.dmj/1077468779   DOI
5 A. E. Brouwer and W. H. Haemers, Spectra of Graphs, Universitext, Springer, New York, 2012. https://doi.org/10.1007/978-1-4614-1939-6   DOI
6 E. Ghorbani and H. R. Maimani, On eigensharp and almost eigensharp graphs, Linear Algebra Appl. 429 (2008), no. 11-12, 2746-2753. https://doi.org/10.1016/j.laa.2008.05.005   DOI
7 D. A. Gregory, V. L. Watts, and B. L. Shader, Biclique decompositions and Hermitian rank, Linear Algebra Appl. 292 (1999), no. 1-3, 267-280. https://doi.org/10.1016/S0024-3795(99)00042-7   DOI
8 R. Hammack, W. Imrich, and S. Klavzar, Handbook of Product Graphs, second edition, Discrete Mathematics and its Applications (Boca Raton), CRC Press, Boca Raton, FL, 2011.
9 F. Harary, Graph Theory, Addison-Wesley Publishing Co., Reading, MA, 1969.
10 T. Kratzke, B. Reznick, and D. West, Eigensharp graphs: decomposition into complete bipartite subgraphs, Trans. Amer. Math. Soc. 308 (1988), no. 2, 637-653. https://doi.org/10.2307/2001095   DOI