Graphs can be represented using the adjacency matrix $ A \in {\rm I\!R} ^{|\mathcal{V}| \times  |\mathcal{V}| }$

Each node in the graph has a row and a column in the adjacency matrix.

The presence of edges is  represented as an entry in the matrix as  $ A[u,v] = 1 $ if $ (u,v) \in \mathcal{E}$ and $ A[u,v] = 0 $ otherwise


Representation of Graphs

 - If edges are undirected then the adjacency matrix is a symmetric matrix.
 - If edges are directed then the adjacency matrix need not be a symmetric matrix.
 - Some graphs have weighted edges, so the entries in the adjacency matrix are real values rather than 0 or 1.


Properties of Edges in a Graph Data-structure

A graph $$\mathcal{G} = ( \mathcal{V},\mathcal{E})$$ is a set of nodes and edges. An edge going from node $$u \in \mathcal{V}$$ to node $$v \in \mathcal{V}$$ is denoted as $$(u,v) \in \mathcal{E}$$.

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Graph is a general data structure  to represent complex data in complex systems. 
It is used to describe and analyze objects/entities with relations/interactions.
A graph is a collection of nodes( objects ) and edges (interactions)


Graph Data structure

Hamilton, W.L. Graph Representation Learning. *Synthesis Lectures on Artificial Intelligence and Machine Learning*. 2020, 14(3), 1–159. https://doi.org/10.2200/S01045ED1V01Y202009AIM046

Graph Representation Learning by William Hamilton

 - Computer networks
 - Social networks data
 - Economic network data
 - Biomedical network
 - Communication networks
 - Internet
 - Network of neurons
 - Citation between different papers
 - Knowledge graphs



Examples of Data Represented in Graph Data-structure


Multi-relational graphs have different types of edges. The edge notation can be rewritten as $$(u, \tau, v) \in \mathcal{E}$$, where $$\tau$$ is the relation type. An adjacency matrix $$A_{\tau}$$ has to be defined for each edge type. The entire graph is summarized by an adjacency tensor $$\mathcal{A} \in \mathbb{R}^{|\mathcal{V}| \times |\mathcal{R}| \times |\mathcal{V}|}$$, where $$\mathcal{R}$$ is the set of relations. Two important subsets of multi-relational graphs are:
- Heterogeneous graphs
- Multiplex graphs

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