For example: Suppose you want to store a directed graph in your database². You'd use a table to hold your node information, like so³:

create table nodes (
    node_id int primary key,
    name varchar(32)
    -- other node info goes here
)
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Next, you create a "many-to-many" table to describe your edges:

create table edges (
    from_node_id int not null references nodes(node_id),
    to_node_id int not null   references nodes(node_id),
    description varchar(32)
    -- other edge info goes here
)
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Now you can create a graph that's easy to query using normal SQL, so you don't have to slurp all the data to your local box and then reprocess it. Quickie example:

-- Allowed states for "Daily Work Task"
insert graph_nodes values (1, 'Wait/Idle')
insert graph_nodes values (2, 'Ready')
insert graph_nodes values (3, 'Running')
insert graph_nodes values (4, 'Completed')
insert graph_nodes values (5, 'Fault')
insert graph_nodes values (6, 'Disabled')

-- Normal execution path
insert graph_edges values (1, 2, 'All prerequisites complete')
insert graph_edges values (2, 3, 'User makes request')
insert graph_edges values (3, 4, 'Job completes successfully')
insert graph_edges values (4, 1, 'NewDay resets us at midnight')

-- Abnormal paths
insert graph_edges values (3, 5, 'Job faults')
insert graph_edges values (5, 6, 'Manual intervention required')
insert graph_edges values (6, 1, 'System restarted')
insert graph_edges values (5, 2, 'Fault resolved normally')
insert graph_edges values (1, 6, 'System halted')
insert graph_edges values (2, 6, 'System halted')
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Now you can do some simple queries, letting the database do the work for you:

-- Where can I go from node 5?
select  FR.node_id, '-->', TO.node_id, 'When: ' + E.description
from    edges E
join    nodes FR on FR.node_id=E.from_node_id
join    nodes TO on TO.node_id=E.to_node_id
where   E.from_node_id = 5
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which should yield (untested!):

node_id     node_id
------- --- ------- ----------------------------------
      5 -->       2 When: Fault resolved normally
      5 -->       6 When: Manual intervention required
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Notes:

1: Sure, each edge is a 1-to-1 connection, but from the node's perspective, each node may have 'many' input arcs and 'many' output arcs. So each row of the many-to-many table is a natural description of an arc.

2: If you want a plain (i.e. undirected) graph, then you have two choices: (a) You could rename from_node_id and to_node_id to node_1_id and node_2_id; and modify your queries to take into account that either node ID could be the answer to your query. It's a little more work in your SQL, but not terribly much; or (b) simply add two edges, one from node 1 to node 2, and one from node 2 to node 1. This simplifies your queries at the expense of more database space. (This is what I normally do, as the disk space is easier for me to spend than debugging time....)

3: I'm most familiar with Sybase/MS SQL Server T-SQL syntax, so that's what I'm writing in. You shouldn't have any trouble adapting to Oracle, mySQL, or database of your own choosing.