Laplacian Spectral Properties of Signed Circular Caterpillars

Authors

Maurizio Brunetti, University of Naples `Federico II' ItalyFollow

Abstract

A circular caterpillar of girth n is a graph such that the removal of all pendant vertices yields a cycle C_n of order n. A signed graph is a pair Γ =(G,σ), where G is a simple graph and σ: E(G) →{+1, -1} is the sign function defined on the set E(G) of edges of G. The signed graph Γ is said to be balanced if the number of negatively signed edges in each cycle is even, and it is said to be unbalanced otherwise. We determine some bounds for the first n Laplacian eigenvalues of any signed circular caterpillar. As an application, we prove that each signed spiked triangle (G(3;p,q,r), σ), i.e. a signed circular caterpillar of girth 3 and degree sequence π_p,q,r=(p+2,q+2,r+2,1,… 1), is determined by its Laplacian spectrum up to switching isomorphism. Moreover, in the set of signed spiked triangles of order N, we identify the extremal graphs with respect to the Laplacian spectral radius and the first two Zagreb indices. It turns out that the unbalanced spiked triangle with degree sequence π_N-3,0,0 and the balanced spike triangle (G(3;\hat{p},\hat{q},\hat{r}),+), where each pair in {\hat{p}, \hat{q}, \hat{r} } differs at most by 1, respectively maximizes and minimizes the Laplacian spectral radius and both the Zagreb indices.

Creative Commons License

This work is licensed under a Creative Commons Attribution 4.0 License.

Recommended Citation

Brunetti, Maurizio (2020) "Laplacian Spectral Properties of Signed Circular Caterpillars," Theory and Applications of Graphs: Vol. 7: Iss. 2, Article 1.
DOI: 10.20429/tag.2020.070201
Available at: https://digitalcommons.georgiasouthern.edu/tag/vol7/iss2/1