DefinitionDefinition: Literature-sourcedProof: Not applicableSession-authored (Fable 5 assisted)judge pass (gpt-5.6-terra)audited 2026-08-28

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The adjacency matrix of a finite simple graph

Definition

Let $G = (V, E)$ be a finite simple graph and let $V = {v_{i} : i < n} = {v_{0}, \dots, v_{n - 1}}$ be an ordered listing of its vertices. The adjacency matrix of $G$ in that vertex order is the matrix $A (G) = (a_{i j}) \in {0, 1}^{n \times n}$ defined, for $i, j < n$ , by

$a_{i j} = {\begin{cases} 1, & if {v_{i}, v_{j}} \in E, \\ 0, & if {v_{i}, v_{j}} \notin E . \end{cases}$

Because $G$ is simple, $a_{i i} = 0$ for every $i$ , and because edges are unordered, $a_{i j} = a_{j i}$ for all $i, j$ . Thus $A (G)$ is a symmetric $n \times n$ matrix over $R$ (Finite rectangular matrices over a commutative ring, their entries, rows and columns).

Changing the vertex order conjugates $A (G)$ by a permutation matrix, so the matrix depends on the chosen ordering but the spectral data attached to it later do not.

Depends on

Used by

Adjacency spectrum, spectral radius, and cospectral graphs Definition
The Laplacian matrix of a finite simple graph Definition
The adjacency spectrum is an isomorphism invariant Proposition
The (i,j) entry of A(G)ᵏ counts walks of length k Theorem
The complete bipartite graph K_m,n has adjacency spectrum {√mn,0ᵐ⁺ⁿ⁻²,-√mn} Theorem
The complete graph Kₙ has adjacency spectrum {n-1,(-1)ⁿ⁻¹} Theorem

Dependency tree · two levels

6 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Richard P. Stanley, Enumerative Combinatorics, Volume 1, Section 4.7 (standard reference, not scraped)