Czechoslovak Mathematical Journal, Vol. 63, No. 4, pp. 909-922, 2013

Minimizing Laplacian spectral radius of unicyclic graphs with fixed girth

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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