Bounds on the Q-spread of a graph
Alex Schulte
April 17, 2017
Alex Schulte
Iowa State University
April 17, 2017
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Bounds on the Q-Spread of a Graph
Authors:
Carla Silva Oliveira, School of Statistical Sciences, Rio de Janeiro,
Brazil
Leonardo Silva de Lima, Federal Center of Technological Education,
Rio de Janeiro, Brazil
Nair Maria Maia de Abreu, Federal University of Rio de Janeiro, Brazil
Steve Kirkland, University of Manatoba, Canada
Alex Schulte
Iowa State University
April 17, 2017
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Definitions
Definition
For an n × n complex matrix M, the spread, s (M ) of M is defined as the
diameter of its spectrum. That is,
s (M ) = | λ1 − λn |.
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Iowa State University
April 17, 2017
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Definitions
Definition
For an n × n complex matrix M, the spread, s (M ) of M is defined as the
diameter of its spectrum. That is,
s (M ) = | λ1 − λn |.
Note that for the Laplacian L, since 0 is always the smallest eigenvalue,
studying the spread is equivalent to studying the spectral radius.
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Iowa State University
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Definitions
Definition
For a simple graph G , the signless Laplacian, Q, is the matrix given by
D + A.
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Iowa State University
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Definitions
Definition
For a simple graph G , the signless Laplacian, Q, is the matrix given by
D + A.
We will denote the spectrum of Q as q1 , q2 , . . . , qn such that
q1 ≥ q2 ≥ · · · ≥ qn . Note Q is positive semi-definite, so qn ≥ 0.
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Iowa State University
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Bipartite Result
Theorem (Cvetković et al.)
For a graph G with signless Laplacian Q, qn = 0 if and only if G is
bipartite.
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Iowa State University
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Definitions
Definition
The Q-spread of an n × n complex matrix M is |q1 − qn |, this is denoted
sQ ( G ) .
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Iowa State University
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Definitions
Definition
The Q-spread of an n × n complex matrix M is |q1 − qn |, this is denoted
sQ ( G ) .
Therefore, if G is bipartite, studying the Q-spread of G is equivalent to
studying the spectral radius of Q.
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Iowa State University
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Example
Let P be the Petersen graph.
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Iowa State University
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Example
Let P be the Petersen graph.
The spectrum of Q for P is {6, 4, 4, 4, 4, 4, 1, 1, 1, 1}.
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Iowa State University
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Example
Let P be the Petersen graph.
The spectrum of Q for P is {6, 4, 4, 4, 4, 4, 1, 1, 1, 1}.
Then sQ (P ) = |6 − 1| = 5.
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Iowa State University
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Definition
Note if G has no edges then
q1 = qn = 0 = s Q ( G ) .
For the remainder of this presentation suppose G is a graph on n vertices
and G has at least one edge.
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Iowa State University
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Definition
Note if G has no edges then
q1 = qn = 0 = s Q ( G ) .
For the remainder of this presentation suppose G is a graph on n vertices
and G has at least one edge.
Let On denote the empty graph on n vertices.
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Iowa State University
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Definition
Note if G has no edges then
q1 = qn = 0 = s Q ( G ) .
For the remainder of this presentation suppose G is a graph on n vertices
and G has at least one edge.
Let On denote the empty graph on n vertices.
Let GR (n, k ) denote the join of Kk with On−k .
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Iowa State University
April 17, 2017
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Lowerbounds
Proposition
If G is a graph then sQ (G ) ≥ 2. Moreover sQ (G ) = 2 if and only if G is a
union of disjoint copies of K1 and K2 .
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Iowa State University
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Lowerbounds
Proposition
If G is a graph then sQ (G ) ≥ 2. Moreover sQ (G ) = 2 if and only if G is a
union of disjoint copies of K1 and K2 .
Proposition
If G is a connected bipartite graph then
π
sQ (Pn ) = 2 + 2 cos
≤ sQ ( G ) .
n
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Lowerbounds for qn
Lemma
Let G 6= Kn such that G has k vertices of degree n − 1, then
qn (G ) ≥
Alex Schulte
1
(n + 2k − 2 −
2
q
(n + 2k − 2)2 − 8k (k − 1)).
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April 17, 2017
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Lowerbounds for qn
Lemma
Let G 6= Kn such that G has k vertices of degree n − 1, then
qn (G ) ≥
1
(n + 2k − 2 −
2
q
(n + 2k − 2)2 − 8k (k − 1)).
Corollary
GR (n, k ) is the unique connected graph on n vertices which minimizes qn
over all non-complete graphs with k vertices of degree n − 1.
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Iowa State University
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Upperbounds
Lemma
Let G be connected with ∆(G ) ≤ n − 2, then
sQ (G ) ≤ 2n − 4.
Equality holds when G = C4 .
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Iowa State University
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Upperbounds
Lemma
Let G be connected with ∆(G ) ≤ n − 2, then
sQ (G ) ≤ 2n − 4.
Equality holds when G = C4 .
Lemma
If n ≥ 5 and G has k ≥ 3 vertices of degree n − 1, then
sQ (G ) < 2n − 4.
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Iowa State University
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More Lemmas
Lemma
Let n ≥ 7 and k = 1 or k = 2 and suppose G has no vertices of degree
n − 2, then
q1 (G ) < 2n − 4.
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Iowa State University
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More Lemmas
Lemma
Let n ≥ 7 and k = 1 or k = 2 and suppose G has no vertices of degree
n − 2, then
q1 (G ) < 2n − 4.
Lemma
Let n ≥ 7 and k = 1 or k = 2 and suppose G has at least one vertex of
degree n − 2, then
2k
qn (G ) ≥
.
n−2
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Iowa State University
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Upperbound Result
Theorem
Let G be a connected graph on n ≥ 5 vertices, then
sQ (G ) < 2n − 4.
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Proof
If G = Kn then
sQ (Kn ) = n < 2n − 4.
Therefore we may assume G 6= Kn .
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Proof
If ∆(G ) ≤ n − 2 the result follows from the lemma:
Lemma
Let G be connected with ∆(G ) ≤ n − 2, then
sQ (G ) ≤ 2n − 4.
Equality holds when G = C4 .
Therefore assume ∆(G ) = n − 1.
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Iowa State University
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Proof
If k ≥ 3 the result follows from an earlier lemma:
Lemma
If n ≥ 5 and G has k ≥ 3 vertices of degree n − 1, then
sQ (G ) < 2n − 4.
Therefore k = 1 or k = 2.
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Iowa State University
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Proof
If n ≥ 7 and k = 1, 2 and G has no vertices of degree n − 2 then the
result holds by a previous lemma:
Lemma
Let n ≥ 7 and k = 1 or k = 2 and suppose G has no vertices of degree
n − 2, then
q1 (G ) < 2n − 4.
Therefore we assume G has at least one vertex of degree n − 2.
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Iowa State University
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Proof
If k = 1, 2 and G has a vertex of degree n − 2, then the result holds by an
earlier lemma:
Lemma
Let n ≥ 7 and k = 1 or k = 2 and suppose G has at least one vertex of
degree n − 2, then
qn (G ) ≥
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2k
.
n−2
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Proof
If k = 1, 2 and G has a vertex of degree n − 2, then the result holds by an
earlier lemma:
Lemma
Let n ≥ 7 and k = 1 or k = 2 and suppose G has at least one vertex of
degree n − 2, then
qn (G ) ≥
2k
.
n−2
The remaining cases where n = 5, 6 is done by direct computation.
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Iowa State University
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Result
Hence the theorem holds.
Theorem
Let G be a connected graph on n ≥ 5 vertices, then
sQ (G ) < 2n − 4.
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n ≥ 5 IS Required
If n = 4, then
2n − 4 = 2(4) − 4 = 8 − 4 = 4.
Path, sQ (P4 ) = 2
Star, sQ (K1,3 ) = 4
Cycle, sQ (C4 ) = 4
Complete, sQ (K4 ) = 4
Diamond, sQ (D ) = 4.472 . . .
Paw, sQ (P ) = 4.561 . . . .
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Iowa State University
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Result
Corollary
Let G be a graph on n ≥ 5 vertices, then
sQ (G ) ≤ 2n − 4.
Equality holds if and only if G = Kn−1 ∪ K1 .
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Iowa State University
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Switching Gears
Definition
Let G be a graph on n vertices where
G = K1,n−p −1 ∪ K1 ∪ K1 ∪ · · · ∪ K1 .
Then G is a special case of the path complete graph denoted PCn,p,1 .
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Iowa State University
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Results
Theorem
Let n ≥ 4√and G = PC n,p,1 . Then, the spectrum of PCn,p,1 is
2n+p −4± 4n2 +n(−4p −16)+p 2 +16p +16
(
, ( n − 2 ) (p ) , ( n − 3 ) (n −p −2) )
2
and the Q-spread of G is
sQ (PC n,p ,1 ) =
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q
4n2 + n (−4p − 16) + p 2 + 16p + 16.
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Results
Theorem
Let n ≥ 4√and G = PC n,p,1 . Then, the spectrum of PCn,p,1 is
2n+p −4± 4n2 +n(−4p −16)+p 2 +16p +16
(
, ( n − 2 ) (p ) , ( n − 3 ) (n −p −2) )
2
and the Q-spread of G is
sQ (PC n,p ,1 ) =
q
4n2 + n (−4p − 16) + p 2 + 16p + 16.
Proposition
For n ≥ 7 and 2 ≤ p ≤ n − 2, we have
sQ (PC n,p +1,1 ) < sQ (PC n,p,1 ) ≤ sQ (PC n,1,1 ) =
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p
4n2 − 20n + 33.
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Conjectures
Conjecture
For any connected graph G with n ≥ 5 vertices,
p
sQ (G ) ≤ 4n2 − 20n + 33.
The upper bound is attained if and only if G = PCn,1,1 .
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Conjectures
Conjecture
For any connected graph G with n ≥ 5 vertices,
p
sQ (G ) ≤ 4n2 − 20n + 33.
The upper bound is attained if and only if G = PCn,1,1 .
The conjecture above is equivalent to:
Conjecture
There is no connected graph G with n ≥ 5 vertices such that
p
4n2 − 20n + 33 < sQ (G ) < 2n − 4.
Alex Schulte
Iowa State University
April 17, 2017
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Thank You!
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Iowa State University
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References
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Applied Mathematics Letters (2009), to appear.
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Neighborhood Sarch for Extremal Graphs XI.: Bounds on Algebraic
Connectivity, Graph Theory and Combinatorial Optimization, ed.
Springer, (2005), 1–16.
G. Caporossi and P. Hansen, Variable neighborhood search for
extremal graphs: the AutographiX system, Discrete Mathematics,
212, (2000), 29–34.
D. Cvetković, Signless Laplacians and line graphs, Bulletin T. CXXXI
de l’ Académie serbe des sciences et des arts (2005) Classe des
Sciences mathématiques et naturelles Sciences mathématiques, 30,
(2005), 85–92.
D. Cvetković, P. Rowlinson and S. Simić, Signless Laplacian of finite
graphs, Linear Algebra and its Applications, 423, (2007), 155–171.
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Iowa State University
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