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Let $ G=(V(G),E(G)) $ be a simple undirected graph with vertex set $ V(G)=\{v_{1}, v_{2},\dots, \\v_{n} \} $, and edge set $ E(G) $. If the vertex $ v_{i} $ is adjacent to the vertex $ v_{j} $, then we write $ v_{i} \sim v_{j}$. The undirected edge between the vertices $ v_{i} $ and $ v_{j} $, if exists, is denoted by $ e_{ij} $. The *adjacency matrix* of a simple graph $ G $, denoted by $ A(G) $, is the symmetric $ n \times n $ matrix whose $ (i,j)th $ entry is defined by $ a_{ij}=1 $ if $ v_{i} \sim v_{j} $, and $ a_{ij} = 0 $ otherwise .

A digraph is said to be an *oriented graph* if it has no digons. A *mixed graph* is a graph which may contain both directed and undirected edges. When we consider Hermitian adjacency matrix of a mixed graph, the undirected edges are treated as digons. From this point of view, digraphs and mixed graphs are equivalent.

Suppose $ V(T)=\{v_{1}, v_{2}, \dots, v_{n} \} $ is the vertex set of a tree $ T $. Then for $ 0\leq i , j \leq n $, 