# Minimizing Laplacian spectral radius of unicyclic graphs with fixed girth

### Kamal Lochan Patra NISER, Bhubaneswar, India

#### Binod Kumar Sahoo NISER, Bhubaneswar, India

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Minisymposium: SPECTRAL GRAPH THEORY

Content: We consider the following problem: Over the class of all simple connected unicyclic graphs on \$n\$ vertices with girth \$g\$ (\$n,g\$ being fixed), which graph minimizes the Laplacian spectral radius? Let \$U_{n,g}\$ be the lollipop graph obtained by appending a pendent vertex of a path on \$n-g\;(n> g)\$ vertices to a vertex of a cycle on \$g\geq 3\$ vertices. We prove that the graph \$U_{n,g}\$ uniquely minimizes the Laplacian spectral radius for \$n\geq 2g-1\$ when \$g\$ is even and for \$n\geq 3g-1\$ when \$g\$ is odd.

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