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Bounds of Eigenvalues of
-Minor Free Graphs
Journal of Inequalities and Applications volume 2009, Article number: 852406 (2009)
Abstract
The spectral radius of a graph
is the largest eigenvalue of its adjacency matrix. Let
be the smallest eigenvalue of
. In this paper, we have described the
-minor free graphs and showed that (A) let
be a simple graph with order
. If
has no
-minor, then
. (B) Let
be a simple connected graph with order
. If
has no
-minor, then
, where equality holds if and only if
is isomorphic to
.
1. Introduction
In this paper, all graphs are finite undirected graphs without loops and multiple edges. Let be a graph with
vertices,
edges, and minimum degree
or
. The spectral radius
of
is the largest eigenvalue of its adjacency matrix. Let
be the smallest eigenvalue of
. The join
is the graph obtained from
by joining each vertex of
to each vertex of
. A graph
is said to be a minor of
if
can be obtained from
by deleting edges, contracting edges, and deleting isolated vertices. A graph
is
-minor free if
has no
-minor.
Brualdi and Hoffman [1] showed that the spectral radius satisfies , where
, with equality if and only if
is isomorphic to the disjoint union of the complete graph
and isolated vertices. Stanley [2] improved the above result. Hong et al. [3] showed that if
is a simple connected graph then
with equality if and only if
is either a regular graph or a bidegreed graph in which each vertex is of degree either
or
. Hong [4] showed that if
is a
-minor free graph then (1)
, where equality holds if and only if
is isomorphic to
; (2)
, where equality holds if and only if
is isomorphic to
.
In this paper, we have described the -minor free graphs and obtained that
(a)let be a simple graph with order
. If
has no
-minor, then
;
(b)let be a simple connected graph with order
. If
has no
-minor, then
, where equality holds if and only if
is isomorphic to
.
2.
-Minor Free Graphs
The intersection of
and
is the graph with vertex set
and edge set
. Suppose
is a connected graph and
be a minimal separating vertex set of
. Then we can write
, where
and
are connected and
. Now suppose further that
is a complete graph. We say that
is a
-sum of
and
, denoted by
, if
. In particular, let
denote a
sum of
and
. Moreover, if
or
(say
) has a separating vertex set which induces a complete graph, then we can write
such that
and
are connected and
is a complete subgraph of
. We proceed like this until none of the resulting subgraphs
has a complete separating subgraph. The graphs
are called the simplical summands of
. It is easy to show that the subgraphs
are independent of the order in which the decomposition is carried out (see [5]).
Theorem 2.1 (see [6], D. W. Hall; K. Wagner).
A graph has no -minor if and only if it can be obtained by
-,
-,
-summing starting from planar graphs and
.
A graph is said to be a edge-maximal
-minor free graph if
has no
-minor and
has at least an
-minor, where
is obtained from
by joining any two nonadjacent vertices of
. A graph
is called a maximal planar graph if the planarity will be not held by joining any two nonadjacent vertices of
.
Corollary 2.2.
If is an edge maximal
-minor free graph then it can be obtained by
-summing starting from
and edge maximal planar graphs.
Proof.
This follows from Theorem 2.1.
Lemma 2.3.
If and
are two maximal planar graphs with order
and
, respectively, then
is not a maximal planar graph.
Proof.
We denote a planar embedding of by
still. Since
is a maximal planar graph, every face boundary in
is a 3-cycle. Hence the outside face boundary in
is a 4-cycle, this implies that the graph
is not maximal planar.
Further, we have the following results.
Theorem 2.4.
If is an edge-maximal
-minor free graph with
vertices then
, where
,
is a maximal planar graph with order
.In particular,
(1)when , where
;
(2)when , where
;
(3)when , where
;
(4)when is a maximal planar graph.
Proof.
Suppose that the graphs are the simplical summands of
, namely
. By Corollary 2.2,
is either a maximal planar graph or a
. By Lemma 2.3, there is at most a maximal planar graph in
. Hence we have
, where
,
is a maximal planar graph with order
.
Lemma 2.5 (see [7]).
Let be a simple planar bipartite graph with
vertices and
edges. Then
.
Theorem 2.6.
Let be a simple connected bipartite graph with
vertices and
edges. If
has no
-minor, then
.
Proof.
Let be a simple connected edge-maximal
-minor free graph with
vertices and
edges. Suppose that the graphs
are the simplical summands of
. Then
is either a maximal planar graph or the graph
by Corollary 2.2. Further, without loss generality, we may assume that
is a spanning subgraph of
. Let the graph
be the intersection of
and
. Then
for
. If
then
is a subgraph of
, implies that
. If
is a maximal planar graph then
is a simple planar bipartite graph, implies that
by Lemma 2.5. Next we prove this result by induction on
. For
,
. Now we assume it is true for
and prove it for
. Let
and
. Then
by the induction hypothesis.
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F852406/MediaObjects/13660_2009_Article_2018_IEq201_HTML.gif)
. Hence .
3. Bounds of Eigenvalues of
-Minor Free Graphs
Lemma 3.1 (see [3]).
If is a simple connected graph then
with equality if and only if
is either a regular graph or a bidegreed graph in which each vertex is of degree either
or
.
Lemma 3.2.
Let be a simple connected graph with
vertices and
edges. If
,then
, where equality holds if and only if
and
is either a regular graph or a bidegreed graph in which each vertex is of degree either
or
.
Proof.
Because when and
,
is a decreasing function of
for
, this follows from Lemma 3.1.
Lemma 3.3.
Let be a maximal planar graph with order
, and let
be a graph with
vertices and
edges.
(1)If and
, where
, then
.
(2)If and
, where
, then
,
.
(3)If and
, where
, then
,
.
Proof.
Applying the properties of the maximal planar graphs, this follows by calculating.
Lemma 3.4.
Let be a maximal planar graph with order
, and let
be a graph with
vertices.
(1)If and
, where
, then
.
(2)If and
, where
, then
.
(3)If and
, where
, then
.
Proof.
It follows that (1) and (3) are true by Lemma 3.2 and 5(1)( 3). Next we prove that (2) is true too.
Let be a graph obtained from
by expanding
(in the simplcal summands of
) to
, such that
can be obtained by
-summing
, namely,
.
This implies that by (1). Also we have
, so
.
Theorem 3.5.
Let be a simple graph with order
. If
has no
-minor, then
.
Proof.
Since when adding an edge in the spectral radius
is strict increasing, we consider the edge-maximal
-minor free graph only. Next we may assume that
is an edge-maximal
-minor free graph.
By Theorem 2.4 and Lemma 3.4, when ,
.
When ,
.
When , we have
by calculating directly, where
,
is a maximal planar graph with order
(see Theorem 2.4).
Therefore when ,
.
Remark 3.6.
In Theorem 3.5, the equality holds only if , for the others, the upper bounds of
are not sharp. We conjecture that the best bound of
is
still.
Lemma 3.7 (see [7]).
If is a simple connected graph with
vertices, then there exists a connected bipartite subgraph
of
such that
with equality holding if and only if
.
Lemma 3.8 (see [7]).
If is a connected bipartite graph with
vertices and
edges, then
, where equality holds if and only if
is a complete bipartite graph.
Theorem 3.9.
Let be a simple connected graph with
vertices. If
has no
-minor, then
, where equality holds if and only if
is isomorphic to
.
Proof.
This follows from Lemmas 3.7, 3.8 and Theorem 2.6.
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Acknowledgments
The author wishes to express his thanks to the referee for valuable comments which led to an improved version of the paper. Work supported by NNSF of China (no. 10671074) and NSF of Zhejian Province (no. Y7080364).
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Fang, KF. Bounds of Eigenvalues of -Minor Free Graphs.
J Inequal Appl 2009, 852406 (2009). https://doi.org/10.1155/2009/852406
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DOI: https://doi.org/10.1155/2009/852406