node in the graph is an atom and represented by a nonlinear oscillator. These nonlinear oscillators are connected through the graph connectivity. In this manner, one can study protein structure and function through the nonlinear dynamics theory widely used in chaos, synchronization, stability, pattern formation, etc.

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   ![Illustration of a graph. The associate adjacent, weight and Laplacian matrix can be found in Eq. (\[eq:graph\_matrix\]). []{data-label="fig:GRAPH"}](GRAPH.png "fig:"){width="30.00000%"}
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### Elementary graph theory {#sec:Graph}

Graph theory deals with a set of discrete vertices (or atoms) and their connectivity (or bonds). Normally, an undirected graph $G$ can be denoted as a pair $G(V,E)$, where $V= \{v_i;i=1,2,...,N \}$ denotes its set of $N$ vertices (or protein atoms), $N=|V|$. Here $E=\{e_i=(v_{i_1},v_{i_2});1\leq i_1 \leq N, 1\leq i_2 \leq N \}$ denotes its set of edges, which can be understood as certain covalent or noncovalent bonds among atoms in a molecule. Each edge in $E$ is an unordered pair of vertices, with the edge connecting distinct vertices $v_{i_1}$ and $v_{i_2}$ written as $e_i=(v_{i_1},v_{i_2})$. Then the adjacency matrix $A$ of $G$ is given by [@ChungOverview; @Mohar:1991laplacian; @Mohar:1997some; @von:2007tutorial] $$\begin{aligned}
\label{eq:couple_matrix28}
A_{ij}=\begin{cases} \begin{array}{ll}
            1 & (v_i,v_j) \in E\\
            0 & (v_i,v_j) \not \in E.\\
          \end{array}
\end{cases}\end{aligned}$$ The degree of 