This paper investigates the Turan-like problem for -free unbalanced signed graphs, where is the set of unbalanced signed complete graphs with vertices. The maximum number of edges and the maximum index for -free unbalanced signed graphs are given. Moreover, the extremal -free unbalanced signed graphs with the maximum index are characterized.
Keywords: Signed graph, Extremal graph, Clique, Index
1. Introduction
In this paper, write for a signed graph with underlying graph and sign function on the edge set of . Denote by , , , and the vertex set, the edge
set, the number of edges, and the number of negative edges of , respectively.If
two vertices are
joined by an edge, let the quantity be or depending on whether is a positive or a negative edge. The
adjacency matrix of is
then defined by . The
eigenvalues of is that of
, which are denoted by
. In
particular, the largest eigenvalue of is called the index. The spectral
radius of is the largest
absolute value of the eigenvalues of , denoted by . These two coincide when
the absolute values of the eigenvalues of do not exceed its
index.
This paper aims to investigate the Turán-like problem
for -free
unbalanced
signed graphs, where is the set of all the
-vertex unbalanced
signed complete graphs. Before presenting new theorems, we provide an
introductory discussion.
Given a graph , a graph is called -free, if it contains no as a subgraph. A prime problem of
extremal graph theory is to determine the maximum number of edges in an
-vertex-free graph, known as Turán
problem. Early in 1907, Mantel [10] proved that if is an -vertex triangle-free graph, then
,
with equality holding if and only if ,
that is, is an -vertex complete bipartite graph
with two almost equally size partitions. The study of extremal graph
theory as a subject in its own right was formally initiated by
Turán in 1940. Let be an integer and be a complete graph on vertices. In his seminal
papers [14,15], the maximum number of
edges of a -free graph was
determined, and the unique extremal graph was also characterized.
Turán’s graph, denoted by , is the complete -partite graph on vertices which is the result of
partitioning vertices into almost equally sized partitions
and taking all edges connecting two different partition classes (note
that if then ). Denote the number of edges
in Turán’s graph by . Using these
notations, Turán’s Theorem can be stated as follows.
Theorem 1.1. Let be a graph on vertices. If is -free then . Furthermore, equality
holds if and only if .
Turán’s Theorem is a fundamental theorem in extremal
graph theory, providing insights into the structure of extremal graphs.
In addition to Turán’s Theorem, extremal graph theory also
encompasses many other important results and problems, such as the
shortest path problems [1], graph decomposition problems [8], and so on.
Research into these problems not only helps in understanding the
relationship between the global and local structures of graphs, but also
plays a crucial role in solving combinatorial optimization problems,
network design, and information transmission. The readers can refer to
[3] for a
comprehensive overview.
In the past two decades, the spectral version of Turán
problem is paid much attention by many researchers. Below we only
mention the result raised by Nikiforov in 2007[11], which will be used for study
later in this article. The readers can refer to an outstanding paper
[9] to understand
the current research status of such problems.
Theorem 1.2 ([11, Theorem 1]).
Let be a graph on vertices. If is -free then . Furthermore, equality holds if and only if
.
Note that Turán’s graph is both the extremal graph of
number of edges and of spectral radius. Actually, the Turán
problem and the spectral Turán problem are closely related.
See [12] for
more discussions on this topic. Different from aforementioned studies,
we focus on signed graphs, and ask what are the maximum number of edges
and the maximum spectral radius of an -free signed graph of order
, where is a set of
signed graphs. That problem were initially proposed by
Wang, Hou, and Li in their recent research [16], in which they referred to
such problem as the Turán-like problem in the
context of signed graphs.
Denote by the set of all
unbalanced signed complete graphs on vertices. Let be a
-free
unbalanced signed graph obtained from a signed clique on vertex set with one
negative edge and an
all-positive clique on
vertex set by adding positive edges between each vertex of and all but one vertex of . The superscript denotes the number of vertices not adjacent to . It is obvious that the number of
vertices in not adjacent to is not greater than , that is, In addition, Note that, due to the unbalance, if
, then , and if , then Here we
draw the signed graphs when
to illustrate this structure, as shown in Fig. 1, noting that thin solid
lines (resp., thin dashed lines) represent positive edges (resp.,
negative edges), thick solid lines between two vertex sets represent the
connection of all possible positive edges, and the solid circle
represents all-positive clique.
An important feature of signed graphs is the concept of
switching the signature. For a signed graph and , the operation that changes the signs of all
edges between and is called a
switching operation. We say that the signed graphs and obtained by a
switching operation from is
switching equivalent. It is important to observe that switching
equivalent signed graphs have similar adjacency matrices and so have the
same eigenvalues (see [22]).
Now we present some interesting results about the Turán-like
problem.
Theorem 1.3 ([16, Theorem 1.2]). If is a connected -free unbalanced signed
graph of order , then with equality holding if and only if is switching equivalent to (see Figure 1), where and
Figure 1 The signed graphs and
Theorem 1.4 ([16, Theorem 1.3]).
If
is a connected -free unbalanced signed
graph of order , then with equality holding if and only if is switching equivalent to
Remark 1.5. In Theorems 1.3 and 1.4, the
condition of connectivity can be omitted. In fact, for the former
theorem, if the signed extremal graph is disconnected, we can add an
edge between two connected components to obtain a -free unbalanced signed
graph, leading to a contradiction. For the latter theorem, we know that
for a -free
unbalanced signed graph with the maximum spectral radius, its spectral
radius is equal to its index (cf. [16, p. 62]). Therefore, according to
Proposition 3.2 and Lemma 3.3 , which we will prove later, if
the signed spectral extremal graph is disconnected, we can add a
positive edge between two connected components, getting a -free unbalanced signed
graph with larger index, a contradiction.
In this context, for some special , we reframe the Turán-like
problem in signed graphs to connect it with the spectral Turán problem.
Let be the set of balanced signed
complete graphs on vertices.
Note that, if , then the all-negative signed complete
graph is the unique (up to switching equivalence) -free unbalanced signed graph
attaining the maximum spectral radius (see [5, Theorem 3.1]). This is the trivial case.
Thus, we only detect among -free balanced signed
graphs those having the maximum spectral radius. Moreover, balanced
signed graphs are switching equivalent to all-positive signed graphs. We
can only look for the desired signed graphs in all-positive signed
graphs and this problem becomes the spectral Turán problem
for -free graphs, when
considering the spectral radius.
Recently, several researches have explored related issues (see [7,20,19,17]),
and here we recall several results that will be helpful for study later.
Chen and Yuan, in [7], investigated the
Turán-like problem for -free signed graphs, and
their results are as follows:
Theorem 1.6 ([7, Theorem 1.5]).
If is a -free unbalanced signed
graph of order , then
Remark 1.7. Note that all
the -free unbalanced
signed graphs attaining above upper bound are determined in their paper.
Here, we do not list them and observe that the condition can be removed from Theorem 1.6 by consulting
the tables of signed graphs with order at most 6 (cf. [6]).
Theorem 1.8 ([7], Theorem 1.6).
If is a -free unbalanced signed
graph of order , then with equality
holding if and only if is
switching equivalent to .
Now we are in a position to present our results. The following
theorem concerns the maximum of the number of edges.
Theorem 1.9. If is a -free unbalanced signed
graph of order , then Furthermore, if , then the equality holds if and only if ,
where ,
and
when , when .
Below we show that the -free unbalanced signed graph with maximum index is switching
equivalent to .
Theorem 1.10. If is a -free unbalanced signed
graph of order , then with equality holding if and only if is switching equivalent to , where the
number of in the superscript is
.
The proposition below adds some numerical estimates to Theorem 1.10.
Proposition 1.11. The index equals the largest root of the polynomial
and satisfies
In particular, if and only if .
The remainder of this article is organized as follows: in Section 2, we present some fundamental properties
and conclusions that will be used in the sequel. In Section 3 , we provide the proofs of Theorems 1.9,1.10
and Proposition 1.11. In the concluding remarks, we
analyze our results and provide some comments.
2. Preliminaries
In this section we introduce some results which will be useful in the
sequel. The lemma below concerns equitable partitions.
Consider a partition of the set The characteristic matrix of is the matrix whose columns are the characteristic vectors of
Consider a
symmetric matrix of order , with rows and columns are partitioned
according to . The partition of
is equitable if each submatrix
formed by the rows of and the columns of has constant row sums . The matrix is
called the quotient matrix of
with respect to the equitable partition .
Lemma 2.1 ([4, p.30])
The matrix has the following two kinds of
eigenvectors and eigenvalues:
(i) The eigenvectors in the column space of ; the corresponding eigenvalues
coincide with the eigenvalues of ,
(ii) The eigenvectors orthogonal to the columns of ; the corresponding eigenvalues of
remain unchanged if some scalar
multiple of the all-one block is
added to block for each
Next we present a celebrated result of signed graphs, which is an
important method for determining the switching equivalence of two signed
graphs with the same underlying graph.
Lemma 2.2 ([21], Proposition 3.2]).
Two signed graphs on the
same underlying graph are switching equivalent if and only if they have
the same list of balanced cycles.
The balanced clique number of a signed graph , denoted by , is the maximum order
of a balanced clique in .
The following lemma gives an upper bound on the index of a signed graph
in terms of its order and balanced clique number.
Lemma 2.3 ([18], Proposition 5]).
Let be a signed graph of order . Then
We end this section by following lemma which says that the index of a
signed graph is not greater than that of its underlying
graph.
Lemma 2.4 ([5], Theorem 2.1 and Proposition 2.5]).
For a non-empty signed
graph of order , .
Furthermore, , with equality if and only if is balanced and complete.
3. Proofs
This section is devoted to the proofs of Theorems 1.9, 1.10,
and Proposition 1.11. For notations and concepts of
signed graphs undefined here, we refer the reader to [21]. For
introductory material on the matrix theory of signed graphs see the
survey of Zaslavsky [22] and its references. In
particular, let be a signed
graph, and and be disjoint sets of vertices of . We write:
– for the set of vertices of ;
– for the signed
graph induced by , and for ;
– for the number of edges
joining vertices in to vertices
in ;
– for the set of
neighbors of a vertex in , and for ;
– , , and for that is adjacent to by a positive edge, a negative edge,
and non-edge, respectively;
–
for that is
switching equivalent to .
Proof of Theorem 1.9.
From Theorem 1.3, we know that
our assertion holds for . We
will prove our result by induction on . Assume that the assertion is
true for all , and we prove
it for . Let be a -free unbalanced
signed graph with maximum possible number of edges. Note that is -free, and so
First we claim that
contains an unbalanced signed clique of order . Otherwise, by induction hypothesis we
know a
contradiction.
Let be the vertex set of an
unbalanced signed complete graph with order and let be its complement. Since each vertex in
can have at most neighbours in , the number of edges between and is at most . We see that
If , then the
equality holds if and only if the unbalanced -vertex clique contains the negative
edge, is an all-positive
clique, each vertex of is
adjacent to vertices of by positive edges, and the number of
vertices adjacent to both and
is not great than .
Thus, we have proven that the conclusion holds when . Based on the induction hypothesis,
we complete the proof of Theorem 1.9.
The approach for proving Theorem 1.9 is
inspired by the proof of Turán’s Theorem, with the key
difference being that we use induction on here, whereas his proof employs
induction on .
Proof of Proposition 1.11. We give a
vertex partition as ,
, , and . Then the
adjacency matrix of and its quotient matrix
where 0 and j
represent the zero vector and the all-ones vector of appropriate
dimensions, respectively, and and
denote the identity matrix and
all-ones matrix of appropriate orders, respectively. By Lemma 2.1, the eigenvalues of
are that of A and
the other eigenvalues of A remain if we add some scalar
multiple of or
from the blocks equal to
, , , and . Then A and become
The eigenvalues of matrix except the eigenvalues of
are with multiplicity . Then the eigenvalues of are the
eigenvalues of and with multiplicity . Therefore, . By direct calculation, the characteristic
polynomial of the matrix is
Thus is the largest root of . By simple calculations , and so the
equality holds if and only if . So we have , and from Lemma 2.4.
To simplify the proof of Theorem 1.10,
we shall prove several auxiliary results. First, note that according to
Perron-Frobenius theory, there exists a strictly positive eigenvector
corresponding to the index of a simple connected graph . Furthermore, if we add some edges in
, the index of the resulting graph
is larger than that of . These
are incorrect for signed graphs, but we have following
results.
Lemma 3.1 ([13, Lemma 1]).
Let be a signed graph with vertices. Then there exists a signed
graph switching
equivalent to such that
has a
non-negative eigenvector.
Let be an eigenvector corresponding to the index of a signed graph
. The entry corresponds to the
vertex of . So the eigenvalue equation for
reads as follows The following proposition presents a crucial
method investigating the maximum index in signed graphs, which can be
proven based on [13, Theorem 3] or [2, Proposition 2.1]. For the sake of completeness, we
provide a detailed proof here.
Proposition 3.2. Let
be a signed graph with a
non-negative unit eigenvector corresponding to the
index . If we
perform one of the following perturbations in :
(i) Adding some positive edges,
(ii) Removing some negative edges,
(iii) Reversing the signs of some negative edges,
resulting in a new signed graph , then . The equality holds if and only if the
entries of corresponding to the
endpoints of these edges are all zeros.
And if we perform one of the following perturbations in :
(iv) Rotating the positive edge to the non-edge position , where ,
(v) Reversing the sign of the positive edge and the negative edge , where , resulting in a new signed
graph , then . The equality holds if and only if and .
Proof. For (i), we denote by the set
of added positive edges. By Rayleigh Principle, we have
If , then all the equalities hold and so is an eigenvector of corresponding to the
eigenvalue . Take one
positive edge from , say . We will show . Assume that the added positive
edges with one endpoint are
. According to the following eigenvalue equations,
we obtain . By similar analysis as above,
the entries of corresponding to
the endpoints of added positive edges are all zeros.
The proof for (ii) is similar to that for
(i).
Note that changing negative edges to positive is equivalent to
deleting the negative edges and then adding positive edges, so
(iii) follows easily from (i) and
(ii).
For (iv), we have If , then all the equalities hold and so is an eigenvector of corresponding to the
eigenvalue . In view of
the following eigenvalue equations, we have and .
The proof for (v) is similar to that for
(vi) and we omit it. The converse is clear.
Lemma 3.3. Let be a signed graph with a unit
eigenvector corresponding to . If , then has at most zero components.
Proof. Without loss of generality, assume for a
contradiction that . Delete the corresponding vertices from
to obtain a signed graph
. Then by Rayleigh
Principle and Lemma 2.4, a contradiction.
Remark 3.4. Note from Proposition 1.11 that is a -free unbalanced
signed graph with index , for .
Combining this with Proposition 3.2 and Lemma 3.3, we know that if is a -free unbalanced
signed graph with maximum index, then it is connected. Furthermore, if
, then we can find an
eigenvector corresponding to with no zero
components.
Proof of Theorem 1.10.
From Theorem 1.8 we know
that our assertion holds when . Now by induction on , assume the assertion is true for all
and prove it
for .
Let be a signed graph
having the maximum index over all -free
unbalanced signed graphs. In view of Lemma 3.1
and Remark 3.4
we can find with a positive unit eigenvector
corresponding to . Note from Lemma 2.2 that is also a connected
-free
unbalanced signed graph. We will show step by step.
First, since
is unbalanced, there exist at least one negative edge and at least one
negative cycle. Take a negative cycle of the
shortest length from . We claim , otherwise is -free, and so by Theorem 1.4, a
contradiction. Thus, is
a negative triangle on vertices , , and .
Secondly, we say that all the negative edges of are contained in
. Indeed, if there
exists a negative edge not in , by Proposition 3.2 we may delete
it resulting in a –free
unbalanced signed graph with larger index, a contradiction. Therefore,
the number of negative edges of is either or .
We conclude that contains only one
negative edge. Actually, if not, then is a negative triangle with
three negative edges and by Proposition 3.2 we may reverse
signs of two of those, resulting in a -free
unbalanced signed graph with larger index, which leads to a
contradiction.
Next we claim that contains an
unbalanced -vertex
clique with one negative edge , written as . If not, is -free and by
induction hypothesis and Proposition we know that
, a contradiction.
Without loss of generality, suppose that and
, and further that and . Let , and . We claim that , otherwise there exists a vertex satisfying and , and by Proposition 3.2 we can rotate
the positive edge to the
non-edge position , getting a
-free
unbalanced signed graph with larger index than , a contradiction.
We proceed with our proof and establish that . Assume for a contradiction
that . If there exists
such that is all-positive clique , then by we have that each vertex
in is adjacent to and so is an unbalanced clique of order , a contradiction. So there
exist no such that is , and by Proposition 3.2 we may reverse
the signs of and , resulting in a -free
unbalanced signed graph with larger index than , a contradiction.
Then we know that .
Note that each vertex in is
adjacent to at most vertices in . Then we claim that . Otherwise, there exists a
vertex not adjacent to a
vertex , and then we can rotate the positive edge to the non-edge position , resulting a signed graph with
larger index than by Proposition 3.2 and the fact
, a contradiction.
Summing up,
must be a subgraph of . Furthermore, combining Proposition
3.2 and the fact
that the eigenvector is positive,
actually is
.
Thus, we have proven that the conclusion holds when . Based on the induction hypothesis,
we complete the proof of Theorem 1.10.
4. Concluding remarks
In this paper, we study the problems what is the maximum number of
edges and what is the maximum index for -free unbalanced signed
graphs. The condition of unbalance is necessary, otherwise, up to
switching equivalence, the all-positive complete graph is the unique -free signed graph
with , and is
the unique -free
signed graph with . In this time, this problem is trivial.
Our result partly solve the following problem, which called as
the spectral Turán-like problem for -free signed graphs.
To see this, we need introduce some notations and notions.
Problem 4.1. What is the
maximum spectral radius among all -free unbalanced signed
graphs?
Combining Theorem 1.10
and Lemma 2.3, we can drive the following
proposition. Note that the negation of (denoted by ) is obtained by reversing the
sign of each edge in .
Clearly, the eigenvalues of are obtained by reversing the
signs of the eigenvalues of .
Proposition 4.2. If is a -free
unbalanced signed graph of order ,
then , with equality holding if and only if .
Proof.The assertion holds for by Theorem 1.8. Now
assume that and has
the maximum spectral radius among -free unbalanced
signed graphs. Note that is -free and so .
We claim that . If not, then .
Since contains no -vertex balanced clique, then . Using Lemma 2.3, we have which contradicts to .
So in this time, the problem finding the maximum spectral
radius among -free
unbalanced signed graphs becomes finding the maximum index among -free
unbalanced signed graphs. The latter is solved by Theorem 1.10.
Proposition 4.2 solves Problem 4.1 under the restriction
.
The case of is left and seems more challenging for further
study.
References:
R. K. Ahuja, K. Mehlhorn, J. Orlin, and R. E. Tarjan. Faster algorithms for the shortest path problem.
JOURNAL OF THE ACM, 37(2):213–223, 1990.
https://doi.org/10.1145/77600.77615.
S. Akbari, F. Belardo, F. Heydari, M. Maghasedi, and M. Souri. On the largest eigenvalue of signed unicyclic graphs.
Linear Algebra and its Applications, 581:145–162, 2019.
https://doi.org/10.1016/j.laa.2019.06.016.
B. Bollobás.
Extremal Graph Theory. Academic Press, London, 1978.
A. E. Brouwer and W. H. Haemers.
Spectra of Graphs. Springer, New York, 2011.
M. Brunetti and Z. Stanić. Unbalanced signed graphs with extremal spectral radius or index.
Computational and Applied Mathematics, 41(3):118, 2022.
https://doi.org/10.1007/s40314-022-01814-5.
F. C. Bussemaker, P. J. Cameron, J. J. Seidel, and S. V. Tsaranov. Tables of signed graphs.
Technical report, Eut Report 91-WSK-01, Eindhoven, 1991.
V. Nikiforov. Some new results in extremal graph theory. In
Surveys in Combinatorics 2011, volume 392, London Math. Society Lecture Note Ser., pages 141–181, 2011.
https://doi.org/10.1017/CBO9781139004114.005.
Z. Stanić. Perturbations in a signed graph and its index.
Discussiones Mathematicae – Graph Theory, 38(3):841–852, 2018.
https://doi.org/10.7151/dmgt.2035.
P. Turán. On an extremal problem in graph theory.
Középiskolai Matematikai és Fizikai Lapok, 48:436–452, 1941.
https://doi.org/10.1007/BF02760037.
P. Turán. On the theory of graphs.
Colloquium Mathematicum, 3:19–30, 1954.
D. Wang, Y. Hou, and D. Li. Extremal results for -free signed graphs.
Linear Algebra and its Applications, 681:47–65, 2024.
https://doi.org/10.1016/j.laa.2023.10.024.
W. Wang, Z. D. Yan, and J. G. Qian. Eigenvalues and chromatic number of a signed graph.
Linear Algebra and its Applications, 619:137–145, 2021.
https://doi.org/10.1016/j.laa.2021.02.018.
T. Zaslavsky. Matrices in the theory of signed simple graphs. In B. D. Acharya, G. O. H. Katona, and J. Nesetril, editors,
Advances in Discrete Mathematics and Applications: Mysore, pages 207–229. Ramanujan Mathematical Society, Mysore, 2008.