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Let be the characteristic polynomial of the Laplacian matrix of a graph of order . In this paper, we give four transforms on graphs that decrease all Laplacian coefficients and investigate a conjecture A. Ilic and M. Ilic (2009) about the Laplacian coefficients of unicyclic graphs with vertices and pendent vertices. Finally, we determine the graph with the smallest Laplacian-like energy among all the unicyclic graphs with vertices and pendent vertices.

journal of applied mathematics

On the Laplacian Coefficients and Laplacian-Like Energy of Unicyclic Graphs with<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M1"><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:math>Vertices and<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M2"><mml:mrow><mml:mi>m</mml:mi></mml:mrow></mml:math>Pendent Vertices

On the Laplacian Coefficients and Laplacian-Like Energy of Unicyclic Graphs withnVertices andmPendent Vertices