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**On the Laplacian coefficients and Laplacian-like energy of unicyclic graphs with \(n\) vertices and \(m\) pendant vertices.**
*(English)*
Zbl 1264.05082

Summary: Let \(\Phi(G, \lambda) = \det(\lambda I_n - L(G)) = \sum^n_{k=0}(-1)^k c_k(G)\lambda^{n-k}\) be the characteristic polynomial of the Laplacian matrix of a graph \(G\) of order \(n\).

In this paper, we give four transforms on graphs that decrease all Laplacian coefficients \(c_k(G)\) and investigate a conjecture A. Ilić and M. Ilić [Linear Algebra Appl. 431, No. 11, 2195–2202 (2009; Zbl 1194.05089)] about the Laplacian coefficients of unicyclic graphs with \(n\) vertices and \(m\) pendant vertices. Finally, we determine the graph with the smallest Laplacian-like energy among all the unicyclic graphs with \(n\) vertices and \(m\) pendant vertices.

In this paper, we give four transforms on graphs that decrease all Laplacian coefficients \(c_k(G)\) and investigate a conjecture A. Ilić and M. Ilić [Linear Algebra Appl. 431, No. 11, 2195–2202 (2009; Zbl 1194.05089)] about the Laplacian coefficients of unicyclic graphs with \(n\) vertices and \(m\) pendant vertices. Finally, we determine the graph with the smallest Laplacian-like energy among all the unicyclic graphs with \(n\) vertices and \(m\) pendant vertices.

### Keywords:

Laplacian matrix; Wiener index; starlike trees; characteristic polynomial; Laplacian coefficients; unicyclic graphs; smallest Laplacian-like energy### Citations:

Zbl 1194.05089
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\textit{X. Pai} and \textit{S. Liu}, J. Appl. Math. 2012, Article ID 404067, 11 p. (2012; Zbl 1264.05082)

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### References:

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