4.1. Applications to the Minimum Number of Distinct Eigenvalues of a Graph
In this section, applying the tool of the vertex-clique incidence matrix of a graph associated with its edge clique cover, we characterize a few new classes of graphs with .
If
G and
H are graphs then the
Cartesian product of
G and
H denoted by
, is the graph on the vertex set
with
and
adjacent if and only if either
and
and
are adjacent in
H or
and
are adjacent in
G and
. The first statement in the next theorem can also be found in [
7], however, we include a proof here to aid in establishing the second claim.
Theorem 20. Let with . Then, and G has an SSP matrix realization with two distinct eigenvalues.
Proof. Let
where
and
. Then, we have
where
From the structure of
A, we have
. On the other hand,
where
. Hence
and
.
Now, we show that the matrix A has SSP. We need to prove that the only symmetric matrix satisfying , , and is .
From the two equations , , X must have the following form: , where . The equality gives . Also, we have , i.e., .
Hence . Then for . Considering , we have and , and then . Considering for we arrive at for . This means that the row and column sums in are equal to zero. Now, consider where .
Thus, and, consequently, . Hence, the proof is complete. □
Corollary 6. For even n, we have .
Proof. Let and let H be the graph obtained from the complete bipartite graph by removing a perfect matching. Then, by Theorem 20 and Lemma 3, for H or any subgraph of H, . Considering this and that is a subgraph of H, the result is obtained. □
Theorem 21. Let G be a graph obtained from by removing a perfect matching between and a copy of . Then and G has an SSP matrix realization with two distinct eigenvalues.
Proof. Let
where
and
. Considering the fact that
and
are symmetric, we have
where
From the structure of
A, we have
. On the other hand,
where
. This gives
, which proves
.
Now, we show that the matrix A has SSP. We need to prove that the only symmetric matrix satisfying , , and is .
From the two equations
,
,
X must have the following form:
, where
,
and
. The matrix equation
gives
. From (
25) we also have
, i.e.,
, i.e.,
. This gives
, i.e.,
.
Again from (
25), we have
, that is,
, that is,
, i.e.,
Considering a main diagonal entry, say
, in the above matrix equation, we obtain
Considering the
-entry in the above matrix equation, we obtain
. From the above and (
26),
, that is,
. Using the equation
, we arrive at the matrix equation
. Following a similar argument as in the proof of Theorem 20 we obtain
.
Again from (
25), we have
. Since
, we get
, i.e.,
. Considering both the
and
entries from the matrix equation, we arrive at
and
, that is,
, which gives
. □
Corollary 7. Consider the complete bipartite graph by removing a perfect matching. Define a new graph H by adding a copy of to this graph such that each vertex in is adjacent to the corresponding vertex in a copy of . Then, . Moreover, the result holds for any subgraph of H on the same vertex set.
In [
36], the authors studied the problem of graphs requiring property
. A graph
G has
if it contains a path of length
r and every path of length
r is contained in a cycle of length
s. They prove that the smallest integer
m so that every graph on
n vertices with
m edges has
(or each path of length 2 is contained either in a 3-cycle, or a 4-cycle) is
for all
. Using this, it was noted in [
37] that the above equation from [
36] implies that the smallest number of edges required to guarantee that all graphs
G on
n vertices satisfy
is at least
. For small values of
n, it is known that in fact, equality holds in the previous claim. Namely, if at most
edges are removed from the complete graph
with
, then the resulting graph has a matrix realization with two distinct eigenvalues. Along these lines and based on [
37] the following is a natural conjecture:
Conjecture 1. Removing up to edges from does not change the number of distinct eigenvalues of . That is, for any subgraph H of with ,
We confirm Conjecture 1 for
and note that our analysis of the case
differs slightly from [
37]. For this, we need the next few lemmas.
Lemma 8. Let be the tree given in Figure 3. We have and has an SSP matrix realization with two distinct eigenvalues. Proof. Consider the
matrix
as follows:
Using the Gram–Schmidt method we arrive at a column orthonormal matrix
. In this case, we have
. In addition,
and
. This proves that
. Furthermore,
A has SSP (this can be confirmed using SageMath [
38]), and by Lemma 3, the complement of any subgraph of
on the same vertex set also has a matrix realization with two distinct eigenvalues. □
Lemma 9. Let . Then, and has an SSP matrix realization with two distinct eigenvalues.
Proof. Consider the
matrix
corresponding to the labeled graph
G given in
Figure 4 as follows:
. Also
and
. This proves that
. Furthermore,
A has SSP (a computation that can be verified by SageMath [
38]), and by Lemma 3, the complement of any subgraph of
G on the same vertex set also has a matrix realization with two distinct eigenvalues. □
We now verify that Conjecture 1 holds for .
Theorem 22. Removing up to four edges from does not change the number of distinct eigenvalues of , i.e., for any subgraph H of on seven vertices, with we have
Proof. To establish this result, it is sufficient to prove the complement of any graph
H in
Figure 5 has a matrix realization with two distinct eigenvalues. Suppose that the graphs in
Figure 5 are denoted by
for
from left to right in each row. Then, the graphs
for
are the union of complete bipartite graphs with some isolated vertices. By Lemma 4 (2), the complements of these graphs and any subgraph of these graphs have a matrix realization with two distinct eigenvalues. Additionally,
for
and for any subgraph
of
,
by Lemma 8. Moreover,
and for any subgraph
of
,
by Lemma 9. Additionally, from Lemmas 8 and 9 such realizations exist with the SSP. Hence any subgraph of these graphs has a matrix realization with two distinct eigenvalues. To complete the proof, we need to show the complement graph of
has a matrix realization with two distinct eigenvalues with the SSP. To this end, consider the
matrix
as follows:
Using the Gram–Schmidt method we arrive at a column orthonormal matrix
. We have
. In addition,
and
. Hence,
. Furthermore,
A has SSP (a computation that can be verified by SageMath [
38]), and by Lemma 3, the complement of any subgraph of
on the same vertex set also has a matrix realization with two distinct eigenvalues. □
We require the following results to confirm Conjecture 1 for .
Lemma 10. Let , where is the graph on the left given in Figure 6. Then and has an SSP matrix realization with two distinct eigenvalues. Proof. Given
G as assumed it can be shown without too much difficulty that
, where
is the graph on the right given in
Figure 6 minus an edge
e with one endpoint in
and the other endpoint in
with degree three.
Suppose
is a vertex-clique incidence matrix of
, where the blocks
and
are vertex-clique incidence matrices corresponding to graphs
and
, that is,
From (
24) we have
and
. On the other hand, we have
Consider a vertex-clique incidence matrix
as follows:
Then we have
and
. Given
above, the remainder of the proof is devoted to constructing a matrix
so that following (
27) we have
, for some scalar
c. Consider a matrix
so that
where
a is a constant. Suppose the matrix
This with (
28) leads to the following equations:
Solving this system of non-linear equations, we have a candidate matrix : , where , , and .
It is obvious that
and
. Since the matrices
and
have same nonzero eigenvalues, we have
, and then
. Moreover, applying a basic computation from SageMath [
38], we can confirm that
has SSP and this completes the proof. □
By Lemma 10, has an SSP realization with two distinct eigenvalues. Then by Lemma 3, any supergraph on the same vertex set as G has a realization with the same spectrum as A. In particular, . This is stated in the following corollary.
Corollary 8. Let , where is the right graph given in Figure 6. Then, and has an SSP matrix realization with two distinct eigenvalues. Lemma 11. Let , where is obtained from by joining a vertex to any vertex in . Then, and has an SSP matrix realization with two distinct eigenvalues.
Proof. We know that
, where
e is an edge with one endpoint in
and the other in
. Suppose
is a vertex-clique incidence matrix of
, where blocks
and
are vertex-clique incidence matrices corresponding to graphs
and
, that is,
From (
24) we have
and
. On the other hand, we also have the equations in (
27). Now, we consider a vertex-clique incidence matrix
as follows:
Then,
and
. Given
above, the remainder of the proof is devoted to constructing a matrix
so that following (
27) we have
, for some scalar
c. We need to create a matrix
so that
where
a is a constant. Suppose
This with (
29) leads to the following equations:
Solving these non-linear equations we have
. Thus, we have
It is clear that
and
. Since the matrices
and
have same nonzero eigenvalues, we have
, and
. Moreover, applying a basic computation from SageMath [
38], it follows that
has SSP and this completes the proof. □
By Lemma 11, has an SSP realization with two distinct eigenvalues. By Lemma 3, any supergraph on the same set of vertices as G has a matrix realization with the same spectrum as A. Thus, . This is stated in the following corollary.
Corollary 9. Let . Then, and has an SSP matrix realization with two distinct eigenvalues.
Proposition 1. Let , where . Then and has an SSP matrix realization with two distinct eigenvalues.
Proof. We show that the complement of
G has a matrix realization with two distinct eigenvalues with the SSP. Consider
matrix
with rows labeled as given in
Figure 7 for
:
We have
. Also
and
. This proves that
. To verify that
A has SSP, suppose
X is a symmetric matrix such that
,
, and
. Note to verify
it is equivalent to prove that
is symmetric. Now assume that
X has the form:
and
x is a (possibly) nonzero vector of size
. Since
is symmetric, comparing the (1,3) and (3,1) blocks of
we note that
. So if we set
, then
. Comparing the (1,2) and (2,1) blocks of
gives
Hence, it follows that
and
. Finally, comparing the (2,3) and (3,2) blocks of
, we have
From the above equations we deduce that . Substituting the equations , , and into the equation , yields . Assuming , implies an immediate contradiction. Thus , and it follows, based on the analysis above that . Hence A has the SSP. Using the fact that this matrix realization has the SSP together with Lemma 3, it follows that the complement of any subgraph of G on the same vertex set also realizes distinct eigenvalues. □
Lemma 12. Let G be the graph given in Figure 8. Then, and has an SSP matrix realization with two distinct eigenvalues. Proof. We show that the complement graph of
G has a matrix realization with two distinct eigenvalues with the SSP. To do this, first we consider
matrix
M as follows:
We have
. Also
so
. This proves that
. Furthermore,
A has SSP (observed using SageMath [
38]) and by Lemma 3, the complement of any subgraph of
G on the same vertex has a matrix realization having two distinct eigenvalues. □
Now we are in a position to establish that Conjecture 1 holds for .
Theorem 23. Removing up to five edges from does not change the number of distinct eigenvalues of , i.e., for any subgraph H on eight vertices of with ,
Proof. To establish this result, it is sufficient to prove the complement of any graph
H in
Figure 9 has a matrix realization with two distinct eigenvalues. Suppose that the graphs in
Figure 9 are denoted by
for
from left to right in each row. The graphs
for
are the union of complete bipartite graphs with some isolated vertices. By Lemma 4 (2), the complements of these graphs and any subgraph of these graphs have a matrix realization with two distinct eigenvalues. Additionally,
for
and for any subgraph
of
,
by Theorem 20. For
, we have
and for any subgraph
of
,
by Lemma 12. Additionally, from Theorem 20 and Lemma 12 such realizations exist with the SSP. Hence, any subgraph of these graphs has a matrix realization with two distinct eigenvalues.
Further, by Lemma 5, where the graph is connected. If we remove any edges in from the triangle, then the complement of the result graph has at least two distinct eigenvalues by Lemma 4 (2), and if we remove any edges in from out of the triangle, again by Lemma 5, we have that the complement of the result graph has a matrix realization with at least two distinct eigenvalues. We have and the complement of any subgraph of this graph has a matrix realization with two distinct eigenvalues, by Corollary 8. Moreover, , and the complement of any subgraph of this graph also has a matrix realization with two distinct eigenvalues, by Corollary 9. This completes the proof of the theorem. □