This is from my textbook: $A$ is the adjacency matrix of a finite graph. Writing down $A$ requires one to assign some numbering to the vertices. However, the spectrum of the matrix obtained does not depend on the numbering. It is the spectrum of the linear transformation $A$ on the vector space $K^X$ of maps from $X$ to $K$, where $X$ is the vertex set and $K$ is some field such as $\mathbb{R}$ or $\mathbb{C}$.
I don't know why $A$ is a linear transformation on $K^X$. Thank you for your help!
$A$ is a matrix. A matrix represents a linear transformation. Changing the order the vertices are numbered is equivalent to changing the basis you are using to represent the transformation, which does not change the spectrum.
One way to think of this is that you can view the vertices of the graph as a basis for a vector space over $K$ (we would call this vector space $K^{X}$). The vectors in this vector space are weighted sums of graph vertices (just written as $K$-linear combinations of vertices). Then for each vertex $v$, the linear transformation corresponding to $A$ is $$A : v \mapsto \sum_{u \sim v} u.$$ (Here we are using $u \sim v$ to mean $u$ and $v$ are adjacent in the graph.) The motivation for doing this is that it does not require us to have an ordering for the vertices to talk about this transformation or of its' eigenvalues, while to talk about a matrix we would need to specify an order for the vertices.