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Because of the numerous applications to mathematical chemistry and statistical physics, the number of matchings of a molecular graph has been studied extensively not only by mathematicians but also by physicists and chemists. In this paper, we consider the problem on enumeration of matchings of the silicate molecular graphs and derive the exact formula of the number of matchings of two classes of silicate molecular graphs. Particularly, we show that these two classes of silicate molecular graphs have the same matching entropy. [ABSTRACT FROM AUTHOR]

Let G be a vertex-weighted graph with vertex-weighted function ω : V (G) → R + satisfying ω (v i) = x i for each v i ∈ V (G) = { v 1 , v 2 , ... , v n }. The weighted Wiener index of vertex-weighted graph G is defined as W (G ; x 1 , x 2 , ... , x n) = ∑ 1 ≤ i < j ≤ n x i x j d G (v i , v j) , where d G (v i , v j) denotes the distance between v i and v j in G. In this paper, we give a combinatorial explanation of W (G ; x 1 , x 2 , ... , x n) when G is a tree: W (G ; x 1 , x 2 , ... , x n) = m (S (G) , n − 2) ∏ i = 1 n x i , where m (S (G) , n − 2) is the sum of weights of matchings of some weighted subdivision S (G) of G with n − 2 edges. We also give a similar combinatorial explanation of the weighted Kirchhoff index K (G ; x 1 , x 2 , ... , x n) = ∑ 1 ≤ i < j ≤ n x i x j r G (v i , v j) of a unicyclic graph G , where r G (v i , v j) denotes the resistance distance between v i and v j in G. These results generalize some known formulae on the Wiener and Kirchhoff indices. • Give a combinatorial explanation of the weighted Wiener index of vertex weighted graphs. • Give a combinatorial explanation of the weighted Kirchhoff index of unicyclic graphs. • Generalize some known formulae on the Wiener and Kirchhoff indices of graphs. [ABSTRACT FROM AUTHOR]

Counting monomer-dimers, or equivalently, matchings of a graph G , is an interesting but intriguing question with great difficulties in statistical physics. Let Z (G) denote the number of monomer-dimers of G. In this paper, we propose a new combinatorial method to solve this problem in some special kind of graphs R (G ; D ; H a i b i ) , which are obtained from a graph G with edge set E by replacing each edge e = u i v i ∈ E ﹨ D with a graph H a i b i having two root vertices a i and b i by identifying u i with a i and v i with b i for i = 1 , 2 , ... , | E | − | D | , where D is a matching of G , and H a i b i satisfies: Z (H a i b i ) Z (H a i b i − a i − b i) = Z (H a i b i − a i) Z (H a i b i − b i). As an application, we obtain the exact solution of the monomer-diner problem on a fractal scale-free lattice, which answers a question posed by Zhang et al. in 2011. [ABSTRACT FROM AUTHOR]

Mohammadian (J Alg Combin 52 (2020), 33-39) defined the Laplacian matching polynomial L G (x) of a simple graph G and obtained many similar properties to the matching polynomial. In this paper, we extend the definition of L G (x) to weighted graphs (vertex-weighted and edge-weighted graphs). Using the principle of inclusion and exclusion, we prove that the Laplacian matching polynomial of a weighted graph G is equivalent to the matching polynomial of a weighted subdivision S (G) , which results in a combinatorial explanation of the coefficients of the Laplacian matching polynomial. [ABSTRACT FROM AUTHOR]