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Czechoslovak Mathematical Journal, Vol. 63, No. 4, pp. 909-922, 2013
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Minimizing Laplacian spectral radius of unicyclic graphs with fixed girth

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Kamal Lochan Patra, Binod Kumar Sahoo

* Kamal Lochan Patra*, * Binod Kumar Sahoo*, School of Mathematical Sciences, National Institute of Science Education and Research (NISER), P.O. Sainik School, Bhubaneswar, 751005, Odisha, India, e-mail: ` klpatra@niser.ac.in`, ` bksahoo@niser.ac.in`

**Abstract:** In this paper 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 $g\geq3$ vertices. We prove that the graph $U_{n,g}$ uniquely minimizes the Laplacian spectral radius for $n\geq2g-1$ when $g$ is even and for $n\geq3g-1$ when $g$ is odd.

**Keywords:** Laplacian matrix; Laplacian spectral radius; girth; unicyclic graph

**Classification (MSC 2010):** 05C50

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