On the Mixed Adjacency Matrix of a Mixed Graph
Chandrashekar Adiga, B.R. Rakshith, Wasin So
Presentation by: Adam Blumenthal
April 10, 2017
A. Blumenthal
Mixed Adjacency Matris
Outline
1
Definitions
2
Characteristic Polynomials
3
Energy
A. Blumenthal
Mixed Adjacency Matris
Definition (Mixed Graph)
A mixed graph is a order pair G = (V , E ), where
V = {v1 , v2 , . . . , vn } is a vertex set, and edge set
E = {e1 , e2 , . . . , el }, in which some edges are directed and some
are undirected. We will call directed edges arcs.
A. Blumenthal
Mixed Adjacency Matris
Definition (Mixed Graph)
A mixed graph is a order pair G = (V , E ), where
V = {v1 , v2 , . . . , vn } is a vertex set, and edge set
E = {e1 , e2 , . . . , el }, in which some edges are directed and some
are undirected. We will call directed edges arcs.
Definition (Mixed Cycle)
A mixed graph is a mixed cycle if its underlying graph is a cycle.
A. Blumenthal
Mixed Adjacency Matris
Definition (Mixed Graph)
A mixed graph is a order pair G = (V , E ), where
V = {v1 , v2 , . . . , vn } is a vertex set, and edge set
E = {e1 , e2 , . . . , el }, in which some edges are directed and some
are undirected. We will call directed edges arcs.
Definition (Mixed Cycle)
A mixed graph is a mixed cycle if its underlying graph is a cycle.
Definition (Evenly (or Oddly) Oriented)
A mixed cycle with an even number of arcs is said to be evenly
oriented if it has an even number of arcs in the same direction
when traversing the cycle, otherwise it is said to be oddly
oriented.
A. Blumenthal
Mixed Adjacency Matris
Let G be a mixed graph, then the
defined entrywise as


1


1
(M(G ))uv =

−1



0
A. Blumenthal
mixed adjacency matrix M(G ) is
if uv is an edge,
if (u, v ) is an arc,
if (v , u) is an arc,
otherwise
Mixed Adjacency Matris
Let G be a mixed graph, then the
defined entrywise as


1


1
(M(G ))uv =

−1



0
mixed adjacency matrix M(G ) is
if uv is an edge,
if (u, v ) is an arc,
if (v , u) is an arc,
otherwise
2
1
4
3
A. Blumenthal
Mixed Adjacency Matris
Let G be a mixed graph, then the
defined entrywise as


1


1
(M(G ))uv =

−1



0
mixed adjacency matrix M(G ) is
if uv is an edge,
if (u, v ) is an arc,
if (v , u) is an arc,
otherwise
2
1
4


0 −1 1 −1
1 0 −1 0 


1 1
0
0
1 0
0
0
3
A. Blumenthal
Mixed Adjacency Matris
Theorem (Adiga, Rakshith, and So)
Let G be a mixed graph of order n. Then
X
|M(G )| =
(−1)n+e(H) 2po (H) (−2)pe (H) ,
H∈Hi
where Hj is the set subgraphs of order j containing no mixed cycles
with odd number of arcs.
A. Blumenthal
Mixed Adjacency Matris
Theorem (Adiga, Rakshith, and So)
Let G be a mixed graph of order n. Then
X
|M(G )| =
(−1)n+e(H) 2po (H) (−2)pe (H) ,
H∈Hi
where Hj is the set subgraphs of order j containing no mixed cycles
with odd number of arcs.
Proof.
|M(G )| =
X
π∈Sn
sgn(π)
Y
miπ(i)
i
Notice this product is 0 if and only if there is a vertex j such that
vj is not adjacent to vπ(j) . Therefore we have a subgraph of edges,
arcs, and mixed cycles. We recall that a cycle can be traversed in
both directions, so if a spanning subgraph H has c(H) mixed
cycles, there are 2c(H) permutations which are nonzero.
A. Blumenthal
Mixed Adjacency Matris
Theorem (Adiga, Rakshith, and So)
X
|M(G )| =
(−1)n+e(H) 2po (H) (−2)pe (H) ,
H∈Hi
Proof continued.
If H has a component with an odd number of arcs, there exists a
corresponding permutation with opposite sign by traversing the
cycle in the opposite direction. Hence we can only check
decompositions with edges, arcs, and cycles with an even number
of arcs. Since sgn(π) = (−1)n−c(π) , where c(π) is the number of
disjoint cycles of π. For the remaining subgraphs each of the 2c(H)
n+e(H)+pe (H) to the determinant.
permutations π contributes
X (−1)
Therefore |M(G )| =
(−1)n+e(H) 2po (H) (−2)pe (H) .
H∈Hi
A. Blumenthal
Mixed Adjacency Matris
Definition (Characteristic Polynomial)
The characteristic polynomial of a mixed graph G is
n
X
|xI − M(G )| =
ai (G )x n−i , denoted P(G , x).
i=0
A. Blumenthal
Mixed Adjacency Matris
Definition (Characteristic Polynomial)
The characteristic polynomial of a mixed graph G is
n
X
|xI − M(G )| =
ai (G )x n−i , denoted P(G , x).
i=0
Theorem (Adiga, Rakshith, and So)
Let G be a mixed graph of order n with k edges and m arcs. Then
a0 = 1, a1 = 0, a2 = m − k and a3 = 2(To − Te ), where To is the
number of oddly oriented triangles, and Te is the number of evenly
oriented triangles.
A. Blumenthal
Mixed Adjacency Matris
Definition (Characteristic Polynomial)
The characteristic polynomial of a mixed graph G is
n
X
|xI − M(G )| =
ai (G )x n−i , denoted P(G , x).
i=0
Theorem (Adiga, Rakshith, and So)
Let G be a mixed graph of order n with k edges and m arcs. Then
a0 = 1, a1 = 0, a2 = m − k and a3 = 2(To − Te ), where To is the
number of oddly oriented triangles, and Te is the number of evenly
oriented triangles.
Proof.
This follows from the fact that (−1)i ai is the summation of
determinants of all principal i × i submatrices of M(G ).
A. Blumenthal
Mixed Adjacency Matris
Theorem (Adiga, Rakshith, and So)
Let e be an edge or arc uv or (u, v ) respectively
X in G . Then
P(G , x) = P(G − e, x) ∓ P(G − u − v , x) − 2 (−1)∆C P(G − C , x)
C
where
(
1
∆C =
0
if C is oddly oriented,
otherwise
A. Blumenthal
Mixed Adjacency Matris
Theorem (Adiga, Rakshith, and So)
Let e be an edge or arc uv or (u, v ) respectively
X in G . Then
P(G , x) = P(G − e, x) ∓ P(G − u − v , x) − 2 (−1)∆C P(G − C , x)
C
where
(
1
∆C =
0
if C is oddly oriented,
otherwise
Proof.
Idea: We can consider all Hi as the union of three types of
subgraphs, those not containing e, those containing e as its own
component, and those containing e as part of a mixed cycle. This
observation and some expansion of sums quickly resolves into the
desired result.
A. Blumenthal
Mixed Adjacency Matris
Theorem (Adiga, Rakshith, and So)
Let e be an edge or arc uv or (u, v ) respectively
X in G . Then
P(G , x) = P(G − e, x) ∓ P(G − u − v , x) − 2 (−1)∆C P(G − C , x)
C
where
(
1
∆C =
0
if C is oddly oriented,
otherwise
Proof.
Idea: We can consider all Hi as the union of three types of
subgraphs, those not containing e, those containing e as its own
component, and those containing e as part of a mixed cycle. This
observation and some expansion of sums quickly resolves into the
desired result.
This allows us to find the characteristic polynomial of the Mixed
Adjacency Matrix based on the removal of an edge/arc! How nice!
A. Blumenthal
Mixed Adjacency Matris
Theorem (Adiga, Rakshith, and So)
Let u be a vertex of G . Then
X
P(G , x) = xP(G − u, x) −
P(G − u − v , x)
v
X
X
+
P(G − u − w , x) − 2
(−1)∆C P(G − C , x),
w
C ,u∈C
where uv ∈ E (G ), (u, w ) or (w , u) ∈ E (G ), and C has an even
number of arcs.
A. Blumenthal
Mixed Adjacency Matris
Theorem (Adiga, Rakshith, and So)
Let u be a vertex of G . Then
X
P(G , x) = xP(G − u, x) −
P(G − u − v , x)
v
X
X
+
P(G − u − w , x) − 2
(−1)∆C P(G − C , x),
w
C ,u∈C
where uv ∈ E (G ), (u, w ) or (w , u) ∈ E (G ), and C has an even
number of arcs.
So we can find the characteristic polynomial of the Mixed
Adjacency Matrix by either deleting edges or vertices! How nice!
A. Blumenthal
Mixed Adjacency Matris
Definition
Let G be a mixed graph. The energy of M(G ),
n
X
(G ) =
|λi (G )|, where {λi }ni=1 are the eigenvalues of M(G ).
i=1
A. Blumenthal
Mixed Adjacency Matris
Definition
Let G be a mixed graph. The energy of M(G ),
n
X
(G ) =
|λi (G )|, where {λi }ni=1 are the eigenvalues of M(G ).
i=1
Theorem (Adiga, Rakshith, and So)
Let G be a mixed graph and let k and m be the number of edges
and arcs respectively in G . Then
n
X
λ2i (G ) = 2(k − m).
i=1
Proof.
A. Blumenthal
Mixed Adjacency Matris
Definition
Let G be a mixed graph. The energy of M(G ),
n
X
(G ) =
|λi (G )|, where {λi }ni=1 are the eigenvalues of M(G ).
i=1
Theorem (Adiga, Rakshith, and So)
Let G be a mixed graph and let k and m be the number of edges
and arcs respectively in G . Then
n
X
λ2i (G ) = 2(k − m).
i=1
Proof.
Exercise.
A. Blumenthal
Mixed Adjacency Matris
Definition
Let G be a mixed graph. The energy of M(G ),
n
X
(G ) =
|λi (G )|, where {λi }ni=1 are the eigenvalues of M(G ).
i=1
Theorem (Adiga, Rakshith, and So)
Let G be a mixed graph and let k and m be the number of edges
and arcs respectively in G . Then
n
X
λ2i (G ) = 2(k − m).
i=1
Proof.
Exercise.
Hint: M(G ) = A(G ) + S(G ) where A(G ) is the spanning graph
with only edges, and S(G ) is the spanning graph with only arcs
(entries of 1 and −1).
A. Blumenthal
Mixed Adjacency Matris
Theorem (Adiga, Rakshith, and So)
Let G be a mixed graph of order n and let m be the number of
edges and k be the number of arcs in G . Then
p
2 |k − m| ≤ (G ).
A. Blumenthal
Mixed Adjacency Matris
Theorem (Adiga, Rakshith, and So)
Let G be a mixed graph of order n and let m be the number of
edges and k be the number of arcs in G . Then
p
2 |k − m| ≤ (G ).
Proof.
n
n
X
X
X
Since
λi (G ) = 0, we have
λ2i (G ) = −2 λi (G )λj (G ).
i=1
i=1
i<j
Using the Cauchy-Schwarz inequality
and division, we have with
X
the last theorem |k − m| ≤
|λi (G )||λj (G )|. Therefore
i<j
p
(G ) ≥ 2 |k − m|.
A. Blumenthal
Mixed Adjacency Matris
Theorem (Adiga, Rakshith, and So)
Let H be a spanning subgraph of a complete bipartite graph with
bipartition U and W with |U| = m and |W | = n.
A. Blumenthal
Mixed Adjacency Matris
Theorem (Adiga, Rakshith, and So)
Let H be a spanning subgraph of a complete bipartite graph with
bipartition U and W with |U| = m and |W | = n.
Let H̄ σ be the complement graph of H in B together with an
orientation σ such that all arcs are from U to W .
A. Blumenthal
Mixed Adjacency Matris
Theorem (Adiga, Rakshith, and So)
Let H be a spanning subgraph of a complete bipartite graph with
bipartition U and W with |U| = m and |W | = n.
Let H̄ σ be the complement graph of H in B together with an
orientation σ such that all arcs are from U to W .
Then the spectrum of the mixed graph G obtained by combining
H and H̄ σ consists of 0 with multiplicity n + m − 2 and
√
± 2p − mn, where p is the size of the graph H.
A. Blumenthal
Mixed Adjacency Matris
Theorem (Adiga, Rakshith, and So)
1
There exist (proper) mixed graphs whose eigenvalues are all
real.
A. Blumenthal
Mixed Adjacency Matris
Theorem (Adiga, Rakshith, and So)
1
There exist (proper) mixed graphs whose eigenvalues are all
real.
2
There exist (proper) mixed graphs whose eigenvalues are all
purely imaginary.
A. Blumenthal
Mixed Adjacency Matris
Theorem (Adiga, Rakshith, and So)
1
There exist (proper) mixed graphs whose eigenvalues are all
real.
2
There exist (proper) mixed graphs whose eigenvalues are all
purely imaginary.
3
There exist connected mixed graphs with 0 energy.
A. Blumenthal
Mixed Adjacency Matris
Thank you!
A. Blumenthal
Mixed Adjacency Matris