<s>
In	O
mathematics	O
,	O
an	O
incidence	B-Algorithm
matrix	I-Algorithm
is	O
a	O
logical	B-Algorithm
matrix	I-Algorithm
that	O
shows	O
the	O
relationship	O
between	O
two	O
classes	O
of	O
objects	O
,	O
usually	O
called	O
an	O
incidence	B-Algorithm
relation	I-Algorithm
.	O
</s>
<s>
If	O
the	O
first	O
class	O
is	O
X	O
and	O
the	O
second	O
is	O
Y	O
,	O
the	O
matrix	B-Architecture
has	O
one	O
row	O
for	O
each	O
element	O
of	O
X	O
and	O
one	O
column	O
for	O
each	O
element	O
of	O
Y	O
.	O
</s>
<s>
Incidence	B-Algorithm
matrix	I-Algorithm
is	O
a	O
common	O
graph	O
representation	O
in	O
graph	O
theory	O
.	O
</s>
<s>
It	O
is	O
different	O
to	O
an	O
adjacency	B-Algorithm
matrix	I-Algorithm
,	O
which	O
encodes	O
the	O
relation	O
of	O
vertex-vertex	O
pairs	O
.	O
</s>
<s>
For	O
example	O
,	O
the	O
incidence	B-Algorithm
matrix	I-Algorithm
of	O
the	O
undirected	O
graph	O
shown	O
on	O
the	O
right	O
is	O
a	O
matrix	B-Architecture
consisting	O
of	O
4	O
rows	O
(	O
corresponding	O
to	O
the	O
four	O
vertices	O
,	O
1	O
–	O
4	O
)	O
and	O
4	O
columns	O
(	O
corresponding	O
to	O
the	O
four	O
edges	O
,	O
)	O
:	O
</s>
<s>
If	O
we	O
look	O
at	O
the	O
incidence	B-Algorithm
matrix	I-Algorithm
,	O
we	O
see	O
that	O
the	O
sum	O
of	O
each	O
column	O
is	O
equal	O
to	O
2	O
.	O
</s>
<s>
The	O
oriented	B-Algorithm
incidence	I-Algorithm
matrix	I-Algorithm
of	O
an	O
undirected	O
graph	O
is	O
the	O
incidence	B-Algorithm
matrix	I-Algorithm
,	O
in	O
the	O
sense	O
of	O
directed	O
graphs	O
,	O
of	O
any	O
orientation	O
of	O
the	O
graph	O
.	O
</s>
<s>
The	O
oriented	B-Algorithm
incidence	I-Algorithm
matrix	I-Algorithm
is	O
unique	O
up	O
to	O
negation	O
of	O
any	O
of	O
the	O
columns	O
,	O
since	O
negating	O
the	O
entries	O
of	O
a	O
column	O
corresponds	O
to	O
reversing	O
the	O
orientation	O
of	O
an	O
edge	O
.	O
</s>
<s>
The	O
unoriented	B-Algorithm
incidence	I-Algorithm
matrix	I-Algorithm
of	O
a	O
graph	O
G	O
is	O
related	O
to	O
the	O
adjacency	B-Algorithm
matrix	I-Algorithm
of	O
its	O
line	O
graph	O
L(G )	O
by	O
the	O
following	O
theorem	O
:	O
</s>
<s>
where	O
A(L(G )	O
)	O
is	O
the	O
adjacency	B-Algorithm
matrix	I-Algorithm
of	O
the	O
line	O
graph	O
of	O
G	O
,	O
B(G )	O
is	O
the	O
incidence	B-Algorithm
matrix	I-Algorithm
,	O
and	O
Im	O
is	O
the	O
identity	B-Algorithm
matrix	I-Algorithm
of	O
dimension	O
m	O
.	O
</s>
<s>
The	O
integral	O
cycle	O
space	O
of	O
a	O
graph	O
is	O
equal	O
to	O
the	O
null	B-Algorithm
space	I-Algorithm
of	O
its	O
oriented	B-Algorithm
incidence	I-Algorithm
matrix	I-Algorithm
,	O
viewed	O
as	O
a	O
matrix	B-Architecture
over	O
the	O
integers	O
or	O
real	O
or	O
complex	O
numbers	O
.	O
</s>
<s>
The	O
binary	O
cycle	O
space	O
is	O
the	O
null	B-Algorithm
space	I-Algorithm
of	O
its	O
oriented	O
or	O
unoriented	B-Algorithm
incidence	I-Algorithm
matrix	I-Algorithm
,	O
viewed	O
as	O
a	O
matrix	B-Architecture
over	O
the	O
two-element	O
field	O
.	O
</s>
<s>
The	O
incidence	B-Algorithm
matrix	I-Algorithm
of	O
a	O
signed	O
graph	O
is	O
a	O
generalization	O
of	O
the	O
oriented	B-Algorithm
incidence	I-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
It	O
is	O
the	O
incidence	B-Algorithm
matrix	I-Algorithm
of	O
any	O
bidirected	O
graph	O
that	O
orients	O
the	O
given	O
signed	O
graph	O
.	O
</s>
<s>
The	O
line	O
graph	O
and	O
Kirchhoff	B-Algorithm
matrix	I-Algorithm
properties	O
generalize	O
to	O
signed	O
graphs	O
.	O
</s>
<s>
The	O
definitions	O
of	O
incidence	B-Algorithm
matrix	I-Algorithm
apply	O
to	O
graphs	O
with	O
loops	O
and	O
multiple	O
edges	O
.	O
</s>
<s>
The	O
column	O
of	O
an	O
oriented	B-Algorithm
incidence	I-Algorithm
matrix	I-Algorithm
that	O
corresponds	O
to	O
a	O
loop	O
is	O
all	O
zero	O
,	O
unless	O
the	O
graph	O
is	O
signed	O
and	O
the	O
loop	O
is	O
negative	O
;	O
then	O
the	O
column	O
is	O
all	O
zero	O
except	O
for	O
±2	O
in	O
the	O
row	O
of	O
its	O
incident	O
vertex	O
.	O
</s>
<s>
For	O
example	O
,	O
the	O
incidence	B-Algorithm
matrix	I-Algorithm
of	O
the	O
graph	O
to	O
the	O
right	O
is	O
:	O
</s>
<s>
Because	O
the	O
edges	O
of	O
ordinary	O
graphs	O
can	O
only	O
have	O
two	O
vertices	O
(	O
one	O
at	O
each	O
end	O
)	O
,	O
the	O
column	O
of	O
an	O
incidence	B-Algorithm
matrix	I-Algorithm
for	O
graphs	O
can	O
only	O
have	O
two	O
non-zero	O
entries	O
.	O
</s>
<s>
By	O
contrast	O
,	O
a	O
hypergraph	O
can	O
have	O
multiple	O
vertices	O
assigned	O
to	O
one	O
edge	O
;	O
thus	O
,	O
a	O
general	O
matrix	B-Architecture
of	O
non-negative	O
integers	O
describes	O
a	O
hypergraph	O
.	O
</s>
<s>
The	O
incidence	B-Algorithm
matrix	I-Algorithm
of	O
an	O
incidence	O
structure	O
C	O
is	O
a	O
matrix	B-Architecture
B	O
(	O
or	O
its	O
transpose	O
)	O
,	O
where	O
p	O
and	O
q	O
are	O
the	O
number	O
of	O
points	O
and	O
lines	O
respectively	O
,	O
such	O
that	O
if	O
the	O
point	O
pi	O
and	O
line	O
Lj	O
are	O
incident	O
and	O
0	O
otherwise	O
.	O
</s>
<s>
In	O
this	O
case	O
,	O
the	O
incidence	B-Algorithm
matrix	I-Algorithm
is	O
also	O
a	O
biadjacency	O
matrix	B-Architecture
of	O
the	O
Levi	O
graph	O
of	O
the	O
structure	O
.	O
</s>
<s>
As	O
there	O
is	O
a	O
hypergraph	O
for	O
every	O
Levi	O
graph	O
,	O
and	O
vice	O
versa	O
,	O
the	O
incidence	B-Algorithm
matrix	I-Algorithm
of	O
an	O
incidence	O
structure	O
describes	O
a	O
hypergraph	O
.	O
</s>
<s>
In	O
a	O
similar	O
manner	O
,	O
the	O
relationship	O
between	O
cells	O
whose	O
dimensions	O
differ	O
by	O
one	O
in	O
a	O
polytope	O
,	O
can	O
be	O
represented	O
by	O
an	O
incidence	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
The	O
incidence	B-Algorithm
matrix	I-Algorithm
is	O
an	O
important	O
tool	O
in	O
the	O
theory	O
of	O
block	O
designs	O
.	O
</s>
<s>
Considering	O
the	O
blocks	O
as	O
a	O
system	O
of	O
sets	O
,	O
the	O
permanent	O
of	O
the	O
incidence	B-Algorithm
matrix	I-Algorithm
is	O
the	O
number	O
of	O
systems	O
of	O
distinct	O
representatives	O
(	O
SDRs	O
)	O
.	O
</s>
