<s>
More	O
specifically	O
,	O
given	O
a	O
set	O
Π	O
of	O
properties	O
and	O
relations	B-Algorithm
,	O
a	O
Π-representation	O
of	O
some	O
structure	O
X	O
is	O
a	O
structure	O
Y	O
that	O
is	O
the	O
image	O
of	O
X	O
under	O
a	O
homomorphism	O
that	O
preserves	O
Π	O
.	O
</s>
<s>
Perhaps	O
the	O
most	O
well-developed	O
example	O
of	O
this	O
general	O
notion	O
is	O
the	O
subfield	O
of	O
abstract	O
algebra	O
called	O
representation	O
theory	O
,	O
which	O
studies	O
the	O
representing	O
of	O
elements	O
of	O
algebraic	O
structures	O
by	O
linear	B-Architecture
transformations	I-Architecture
of	O
vector	O
spaces	O
.	O
</s>
<s>
(	O
or	O
,	O
more	O
precisely	O
,	O
non-disjointness	B-Algorithm
)	O
is	O
a	O
symmetric	O
relation	O
.	O
</s>
<s>
Representing	O
a	O
graph	O
by	O
such	O
algebraic	O
structures	O
as	O
its	O
adjacency	B-Algorithm
matrix	I-Algorithm
and	O
Laplacian	B-Algorithm
matrix	I-Algorithm
gives	O
rise	O
to	O
the	O
field	O
of	O
spectral	O
graph	O
theory	O
.	O
</s>
<s>
A	O
particularly	O
nice	O
result	O
in	O
this	O
field	O
is	O
the	O
characterization	O
of	O
the	O
planar	O
graphs	O
,	O
as	O
those	O
graphs	O
whose	O
vertex-edge	O
incidence	O
relations	B-Algorithm
are	O
circle	O
orders	O
.	O
</s>
