DMGT

ISSN 1234-3099 (print version)

ISSN 2083-5892 (electronic version)

https://doi.org/10.7151/dmgt

Discussiones Mathematicae Graph Theory

Journal Impact Factor (JIF 2022): 0.7

5-year Journal Impact Factor (2022): 0.7

CiteScore (2022): 1.9

SNIP (2022): 0.902

Discussiones Mathematicae Graph Theory

Article in volume

Authors:

Z. Stanić

Zoran Stanić

University of Belgrade

email: zstanic@matf.bg.ac.rs

Title:

Some properties of the eigenvalues of the net Laplacian matrix of a signed graph

PDF

Source:

Discussiones Mathematicae Graph Theory 42(3) (2022) 893-903

Received: 2019-09-30 , Revised: 2020-03-10 , Accepted: 2020-03-10 , Available online: 2020-03-18 , https://doi.org/10.7151/dmgt.2314

Abstract:

Given a signed graph $\dot{G}$, let $A_{\dot{G}}$ and $D^{\pm}_{\dot{G}}$ denote its standard adjacency matrix and the diagonal matrix of vertex net-degrees, respectively. The net Laplacian matrix of $\dot{G}$ is defined to be $N_{\dot{G}}=D^{\pm}_{\dot{G}}-A_{\dot{G}}$. In this study we give some properties of the eigenvalues of $N_{\dot{G}}$. In particular, we consider their behaviour under some edge perturbations, establish some relations between them and the eigenvalues of the standard Laplacian matrix and give some lower and upper bounds for the largest eigenvalue of $N_{\dot{G}}$.

Keywords:

(net) Laplacian matrix, edge perturbations, largest eigenvalue, net-degree

References:

  1. M. Fiedler, Algebraic connectivity of graphs, Czechoslovak Math. J. 23 (1973) 298–305.
  2. M. Fiedler, A property of eigenvectors of nonnegative symmetric matrices and its application to graph theory, Czechoslovak Math. J. 25 (1975) 619–633.
  3. H. Gao, Z. Ji and T. Hou, Equitable partitions in the controllability of undirected signed graphs, in: Proceedings of IEEE 14th International Conference on Control and Automation (ICCA), (Piscataway NJ, IEEE 2018) 532–537.
  4. S. Martini, M. Egerstedt and A. Bicchi, Controllability analysis of multi-agent systems using relaxed equitable partitions, Int. J. Systems, Control and Communications 2 (2010) 100–121.
    https://doi.org/10.1504/IJSCC.2010.031160
  5. Z. Stanić, Inequalities for Graph Eigenvalues (Cambridge University Press, Cambridge, 2015).
    https://doi.org/10.1017/CBO9781316341308
  6. Z. Stanić, Net Laplacian controllablity of joins of signed graphs, Discrete Appl. Math. 285 (2020) 197–203.
    https://doi.org/10.1016/j.dam.2020.05.011
  7. Z. Stanić, On the spectrum of the net Laplacian matrix of a signed graph, Bull. Math. Soc. Sci. Math. Roumanie 63 (2020) 205–213.
  8. 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 and J. Nešetřil (Ed(s)), (Ramanujan Math. Soc., Mysore 2010) 207–229.
Close