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In this paper we consider the following problem: Over the class of all simple connected unicyclic graphs on vertices with girth (, being fixed), which graph minimizes the Laplacian spectral radius? Let be the lollipop graph obtained by appending a pendent vertex of a path on
vertices to a vertex of a cycle on vertices. We prove that the graph uniquely minimizes the Laplacian spectral radius for when is even and for when is odd.
The Wiener index of a connected graph is defined as the sum of the distances between all unordered pairs of its vertices. We characterize the graphs which extremize the Wiener index among all graphs on vertices with pendant vertices. We also characterize the graph which minimizes the Wiener index over the graphs on vertices with cut-vertices.
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