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Minimizing Laplacian spectral radius of unicyclic graphs with fixed girth

Kamal Lochan PatraBinod Kumar Sahoo — 2013

Czechoslovak Mathematical Journal

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 3 vertices. We prove that the graph U n , g uniquely minimizes the Laplacian spectral radius for n 2 g - 1 when g is even and for n 3 g - 1 when g is odd.

Wiener index of graphs with fixed number of pendant or cut-vertices

Dinesh PandeyKamal Lochan Patra — 2022

Czechoslovak Mathematical Journal

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 n vertices with k pendant vertices. We also characterize the graph which minimizes the Wiener index over the graphs on n vertices with s cut-vertices.

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