Connected signed graphs L-cospectral to signed ∞-graphs

ABSTRACT

A signed graph is a pair $\mathrm{\Gamma }=(G,\sigma )$, where $G=\left(V\right(G),E(G\left)\right)$ is a graph and $\sigma :E\left(G\right)\to \{+1,-1\}$ is the sign function on the edges of G. For a signed graph we consider the Laplacian matrix defined as $L\left(\mathrm{\Gamma }\right)=D\left(G\right)-A\left(\mathrm{\Gamma }\right)$, where $D\left(G\right)$ is the matrix of vertex degrees of G and the (signed) adjacency matrix $A\left(\mathrm{\Gamma }\right)$. A signed ∞-graph consists of two signed cycles with just one vertex in common. In this paper, we study the Laplacian spectral determination problem for the class of signed ∞-graphs, and we identify all connected L-cospectral mates.

KEYWORDS:

• Signed graph
• Laplacian
• ∞-graph
• spectral determination
• bicyclic

AMS CLASSIFICATIONS:

• 05C50
• 05C22

Acknowledgments

The authors are grateful to the unknown referee for the useful remarks which has led to an improvement of the results presentation.

Disclosure statement

No potential conflict of interest was reported by the authors.

ORCID

Francesco Belardo http://orcid.org/0000-0003-4253-2905

Maurizio Brunetti http://orcid.org/0000-0002-2742-1919

Additional information

Funding

The research of this paper was supported by the project ‘SGTACSMC’ of the Università degli Studi di Napoli Federico II (University of Naples ‘Federico II’) and by the Istituto Nazionale di Alta Matematica ‘Francesco Severi’ (INDAM-GNSAGA) (Italy).