# On the eigenspaces of signed line graphs and signed subdivision graphs

### Francesco Belardo University of Primorska

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Minisymposium: SPECTRAL GRAPH THEORY

Content: In this talk we will consider the spectra of signed graphs, namely graphs with signs $\{+,-\}$ mapped on the edges. Recently, given a signed graph $\Gamma$, we have defined the signed line graph $\mathcal{L}(\Gamma)$ and the signed subdivision graph $\mathcal{S}(\Gamma)$ in a way that some well-known formulas on the characteristic polynomials of (unsigned) graphs are extended to signed graphs. Namely, if $\phi(\Gamma,x)$ and $\psi(\Gamma,x)$ denote the adjacency polynomial and the Laplacian polynomial of $\Gamma$, respectively, then $\phi(\mathcal{L}(\Gamma),x) = (x+2)^{m-n}\psi(\Gamma,x+2)$, and $\phi(\mathcal{S}(\Gamma),x) = x^{m-n}\psi(\Gamma,x^2)$. Here we will focus on the relations intercurring among the eigenspaces of the respective classes of signed graphs. This is a joint research with S.K. Simi\'c and I. Sciriha.

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