##  [Adjacency Matrix](/adjacency-matrix-0) 

 Definition

A two-dimensional array representation of a finite graph whose (i,j) entry records the presence and possibly the weight of an edge between node i and node j.

 

 

 

 

 

 





## Principle

Principle

Encode discrete connectivity in a linear-algebraic object so that graph operations become matrix operations.

 

 

 

 

 





## Demonstration

Demonstration

For an undirected simple graph with nodes {1,2,3,4} and edges {1–2,2–3,3–4}, the adjacency matrix A has zeros on the diagonal and ones at positions (1,2),(2,1),(2,3),(3,2),(3,4),(4,3).

 

 

 

 

## Misapplication

Misapplication

Treating the same adjacency matrix as a complete descriptor for a multigraph without extending entries to counts leads to loss of multiplicity information.

 

 

 

 

 





## Consequence

Consequence

Enables use of matrix algebra (powers, products, index calculations) to study walks, connectivity, and combinatorial properties of the graph.

 

 

 

 

## Reversal

Reversal

An incidence matrix instead records node–edge incidences rather than direct node–node connections, shifting focus from adjacency to edge membership.

 

 

 

 

 





## Boundary

Boundary

Defined for finite graphs; for hypergraphs, multilayer networks, or labeled-edge structures the basic binary adjacency must be extended or replaced.

 

 

 

 

 





## Semantic Tension

Semantic Tension

Versus a weighted adjacency: the unqualified adjacency matrix often implies binary entries, while many analyses assume or require real-valued weights.

 

 

 

 

 





## Synthesis

Synthesis

A compact matrix whose pattern of entries is a direct linear-algebraic encoding of which nodes are connected and how strongly in a finite graph.