Geodesic
Unbalanced unicyclic and bicyclic graphs with extremal spectral radius
Czechoslovak Mathematical Journal, Tome 71 (2021) no. 2, pp. 417-433
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A signed graph $\Gamma $ is a graph whose edges are labeled by signs. If $\Gamma $ has $n$ vertices, its spectral radius is the number $\rho (\Gamma ) := \max \{ | \lambda _i(\Gamma ) | \colon 1 \leq i \leq n \}$, where $\lambda _1(\Gamma ) \geq \cdots \geq \lambda _n(\Gamma )$ are the eigenvalues of the signed adjacency matrix $A(\Gamma )$. Here we determine the signed graphs achieving the minimal or the maximal spectral radius in the classes $\frak U_n$ and $\frak B_n$ of unbalanced unicyclic graphs and unbalanced bicyclic graphs, respectively.
A signed graph $\Gamma $ is a graph whose edges are labeled by signs. If $\Gamma $ has $n$ vertices, its spectral radius is the number $\rho (\Gamma ) := \max \{ | \lambda _i(\Gamma ) | \colon 1 \leq i \leq n \}$, where $\lambda _1(\Gamma ) \geq \cdots \geq \lambda _n(\Gamma )$ are the eigenvalues of the signed adjacency matrix $A(\Gamma )$. Here we determine the signed graphs achieving the minimal or the maximal spectral radius in the classes $\frak U_n$ and $\frak B_n$ of unbalanced unicyclic graphs and unbalanced bicyclic graphs, respectively.
DOI : 10.21136/CMJ.2020.0403-19
Classification : 05C22, 05C50
Keywords: signed graph; spectral radius; bicyclic graph 
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Belardo, Francesco; Brunetti, Maurizio; Ciampella, Adriana. Unbalanced unicyclic and bicyclic graphs with extremal spectral radius. Czechoslovak Mathematical Journal, Tome 71 (2021) no. 2, pp. 417-433. doi: 10.21136/CMJ.2020.0403-19

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