On the Laplacian Coefficients of Tricyclic Graphs with Prescribed Matching Number

Let $ \phi (L(G)) = \text{det}(x I−L(G)) = \Sigma_{k=0}^n (−1)^k c_k (G) x^{n−k} $ be the Laplacian characteristic polynomial of $G$. In this paper, we characterize the minimal graphs with the minimum Laplacian coefficients in $ \mathcal{G}_{n,n+2} (i) $ (the set of all tricyclic graphs with fixed order $n$ and matching number $i$). Furthermore, the graphs with the minimal Laplacian-like energy, which is the sum of square roots of all roots on $ \phi (L(G)) $, is also determined in $ \mathcal{G}_{n,n+2} (i) $.