Spectral radius and average 2-degree sequence of a graph
Speaker : Yu-pei Huang
Advisor : Chih-wen Weng
Department of Applied Mathematics, National Chiao Tung University
黃喻培 (Dep. of A. Math., NCTU)
Spectral radius and average 2-degree sequence of a graph
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Average 2-degree sequence
Let Γ=(X, R) denote a finite undirected, connected graph without loops or
multiple edges with vertex set X, edge set R. The spectral radius ρ(Γ) of
Γ is the largest eigenvalue of its adjacency matrix.
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Spectral radius and average 2-degree sequence of a graph
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Average 2-degree sequence
Let Γ=(X, R) denote a finite undirected, connected graph without loops or
multiple edges with vertex set X, edge set R. The spectral radius ρ(Γ) of
Γ is the largest eigenvalue of its adjacency matrix.
∑
For x ∈ X, we define the average 2-degree Mx := y∼x dy /dx , where dx
is the degree of x. Label the vertices of Γ by 1, 2, · · · , n such that
M1 ≥ M2 ≥ · · · ≥ Mn .
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Spectral radius and average 2-degree sequence of a graph
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Average 2-degree sequence
Let Γ=(X, R) denote a finite undirected, connected graph without loops or
multiple edges with vertex set X, edge set R. The spectral radius ρ(Γ) of
Γ is the largest eigenvalue of its adjacency matrix.
∑
For x ∈ X, we define the average 2-degree Mx := y∼x dy /dx , where dx
is the degree of x. Label the vertices of Γ by 1, 2, · · · , n such that
M1 ≥ M2 ≥ · · · ≥ Mn . A graph of order n with identical average 2-degree
(i.e. M1 = M2 = · · · = Mn ) is called pseudo-regular.
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Spectral radius and average 2-degree sequence of a graph
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Non-regular pseudo-regular graphs
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Spectral radius and average 2-degree sequence of a graph
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Non-regular pseudo-regular graphs
surrounded
Surrounded
by
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Spectral radius and average 2-degree sequence of a graph
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Perron-Frobenius Theorem
The following theorem is a fundamental result on the study of Matrix
Theory. It is referred to as Perron-Frobenius Theorem.
.
Theorem
.
If B is a nonnegative irreducible n × n matrix with largest eigenvalue ρ(B)
and row-sums r1 , r2 , . . . , rn , then
ρ(B) ≤ max ri
1≤i≤n
with
equality if and only if the row-sums of B are all equal.
.
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An Application of Perron-Frobenius Theorem


3 1 2
A=1 1 2
1 1 1
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Spectral radius and average 2-degree sequence of a graph
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An Application of Perron-Frobenius Theorem


3 1 2
r1 = 6


A= 1 1 2
r2 = 4
1 1 1
r3 = 3
黃喻培 (Dep. of A. Math., NCTU)
Spectral radius and average 2-degree sequence of a graph
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An Application of Perron-Frobenius Theorem


3 1 2
r1 = 6


A= 1 1 2
r2 = 4
1 1 1
r3 = 3
黃喻培 (Dep. of A. Math., NCTU)
=⇒
ρ(A) ≤ 6.
Spectral radius and average 2-degree sequence of a graph
6 / 18
An Application of Perron-Frobenius Theorem


3 1 2
r1 = 6


A= 1 1 2
r2 = 4
1 1 1
r3 = 3
=⇒
ρ(A) ≤ 6.
Consider AT , ρ(A) ≤ 5.
黃喻培 (Dep. of A. Math., NCTU)
Spectral radius and average 2-degree sequence of a graph
6 / 18
An Application of Perron-Frobenius Theorem


3 1 2
r1 = 6


A= 1 1 2
r2 = 4
1 1 1
r3 = 3
=⇒
ρ(A) ≤ 6.
Consider AT , ρ(A) ≤ 5.
In fact,
√
√
√
e(A) = 1, 2 − 6, 2 + 6, ρ(A) = 2 + 6 + 4.45
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Spectral radius and average 2-degree sequence of a graph
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Upper bounds of spectral radii
.
Theorem
.
Let Γ be a connected graph. Then
ρ(Γ) ≤ M1
.with equality if and only if Γ is pseudo-regular.
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Spectral radius and average 2-degree sequence of a graph
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Upper bounds of spectral radii
.
Theorem
.
Let Γ be a connected graph. Then
ρ(Γ) ≤ M1
.with equality if and only if Γ is pseudo-regular.
.
Proof.
.
.
黃喻培 (Dep. of A. Math., NCTU)
Spectral radius and average 2-degree sequence of a graph
7 / 18
Upper bounds of spectral radii
.
Theorem
.
Let Γ be a connected graph. Then
ρ(Γ) ≤ M1
.with equality if and only if Γ is pseudo-regular.
.
Proof.
.
Let A be the adjacency matrix of a connected graph Γ. Setting
B = (bij ) = U−1 AU, where U = diag (d1 , d2 , · · · , dn ). Let rℓ be the ℓ-th
row sum of B.
.
黃喻培 (Dep. of A. Math., NCTU)
Spectral radius and average 2-degree sequence of a graph
7 / 18
Upper bounds of spectral radii
.
Theorem
.
Let Γ be a connected graph. Then
ρ(Γ) ≤ M1
.with equality if and only if Γ is pseudo-regular.
.
Proof.
.
Let A be the adjacency matrix of a connected graph Γ. Setting
B = (bij ) = U−1 AU, where U = diag (d1 , d2 , · · · , dn ). Let rℓ be the ℓ-th
row sum of B. Then bij = dj aij /di and rℓ = Mℓ .
.
黃喻培 (Dep. of A. Math., NCTU)
Spectral radius and average 2-degree sequence of a graph
7 / 18
Upper bounds of spectral radii
.
Theorem
.
Let Γ be a connected graph. Then
ρ(Γ) ≤ M1
.with equality if and only if Γ is pseudo-regular.
.
Proof.
.
Let A be the adjacency matrix of a connected graph Γ. Setting
B = (bij ) = U−1 AU, where U = diag (d1 , d2 , · · · , dn ). Let rℓ be the ℓ-th
row sum of B. Then bij = dj aij /di and rℓ = Mℓ . Since A and B are similar,
applying Perron-Frobenius Theorem to B, we have ρ(Γ) ≤ M1 with
.equality if and only if Γ is pseudo-regular.
黃喻培 (Dep. of A. Math., NCTU)
Spectral radius and average 2-degree sequence of a graph
7 / 18
Upper bounds of spectral radii
.
Theorem
.
Let Γ be a connected graph. Then
ρ(Γ) ≤ M1
.with equality if and only if Γ is pseudo-regular.
黃喻培 (Dep. of A. Math., NCTU)
Spectral radius and average 2-degree sequence of a graph
8 / 18
Upper bounds of spectral radii
.
Theorem
.
Let Γ be a connected graph. Then
ρ(Γ) ≤ M1
.with equality if and only if Γ is pseudo-regular.
.
Theorem (Chen, Pan and Zhang, 2011)
.
Let Γ be a connected graph. Let a := max {di /dj | 1 ≤ i, j ≤ n}. Then
√
M2 − a + (M2 + a)2 + 4a(M1 − M2 )
ρ(Γ) ≤
,
2
with
equality if and only if Γ is pseudo-regular.
.
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Spectral radius and average 2-degree sequence of a graph
8 / 18
Main result
The following Theorem is our main result which is a generalization of the
previous theorem.
.
Theorem
.
For any b ≥ max {di /dj | i ∼ j} and 1 ≤ ℓ ≤ n,
√
∑
Mℓ − b + (Mℓ + b)2 + 4b ℓ−1
i=1 (Mi − Mℓ )
ρ(Γ) ≤
:= ϕℓ ,
2
with
equality if and only if Γ is pseudo-regular.
.
黃喻培 (Dep. of A. Math., NCTU)
Spectral radius and average 2-degree sequence of a graph
9 / 18
Main result
The following Theorem is our main result which is a generalization of the
previous theorem.
.
Theorem
.
For any b ≥ max {di /dj | i ∼ j} and 1 ≤ ℓ ≤ n,
√
∑
Mℓ − b + (Mℓ + b)2 + 4b ℓ−1
i=1 (Mi − Mℓ )
ρ(Γ) ≤
:= ϕℓ ,
2
with
equality if and only if Γ is pseudo-regular.
.
.
Proof.
.
For each 1 ≤ i ≤ ℓ − 1, let xi ≥ 1 be a variable to be determined later. Let
U = diag(d1 x1 , . . . , dℓ−1 xℓ−1 , dℓ , . . . , dn ) be a diagonal matrix of size n × n.
Consider the matrix B = U−1 AU. Note that A and B have the same
eigenvalues.
Let r1 , r2 , . . . , rn be the row-sums of B.
.
黃喻培 (Dep. of A. Math., NCTU)
Spectral radius and average 2-degree sequence of a graph
9 / 18
.
Proof. (continue)
.
Then for 1 ≤ i ≤ ℓ − 1, let
xi = 1 +
Mi − Mℓ
≥ 1.
ϕℓ + b
we have
ri
ℓ−1
n
∑
∑
1
1
=
aik dk xk +
aik dk
di x i
di x i
k=1
k=ℓ
.
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Spectral radius and average 2-degree sequence of a graph
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.
Proof. (continue)
.
Then for 1 ≤ i ≤ ℓ − 1, let
xi = 1 +
Mi − Mℓ
≥ 1.
ϕℓ + b
we have
ri
ℓ−1
n
∑
∑
1
1
=
aik dk xk +
aik dk
di x i
di x i
=
k=1
ℓ−1
∑
1
xi
k=1
k=ℓ
(xk − 1)aik
n
1 ∑ dk
dk
+
aik
di
xi
di
k=1
.
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Spectral radius and average 2-degree sequence of a graph
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.
Proof. (continue)
.
Then for 1 ≤ i ≤ ℓ − 1, let
xi = 1 +
Mi − Mℓ
≥ 1.
ϕℓ + b
we have
ri
ℓ−1
n
∑
∑
1
1
=
aik dk xk +
aik dk
di x i
di x i
=
≤
k=1
ℓ−1
∑
k=ℓ
1 ∑ dk
dk
+
aik
di
xi
di
k=1
k=1


ℓ−1
1
b ∑
xk − (ℓ − 2) + Mi
xi
xi
1
xi
n
(xk − 1)aik
k=1,k̸=i
.
黃喻培 (Dep. of A. Math., NCTU)
Spectral radius and average 2-degree sequence of a graph
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.
Proof. (continue)
.
Then for 1 ≤ i ≤ ℓ − 1, let
xi = 1 +
Mi − Mℓ
≥ 1.
ϕℓ + b
we have
ri
ℓ−1
n
∑
∑
1
1
=
aik dk xk +
aik dk
di x i
di x i
=
≤
k=1
ℓ−1
∑
k=ℓ
1 ∑ dk
dk
+
aik
di
xi
di
k=1
k=1


ℓ−1
1
b ∑
xk − (ℓ − 2) + Mi
xi
xi
1
xi
n
(xk − 1)aik
k=1,k̸=i
.
= ϕℓ
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Spectral radius and average 2-degree sequence of a graph
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.
Proof. (continue)
.
Similarly for ℓ ≤ j ≤ n we have
rj =
ℓ−1
∑
dk ∑ dk
+
ajk
dj
dj
n
xk ajk
k=1
ℓ−1
∑
k=ℓ
dk ∑ dk
=
(xk − 1)ajk +
ajk
dj
dj
k=1
k=1
( ℓ−1
)
∑
≤ b
xk − (ℓ − 1) + Mℓ
n
k=1
= ϕℓ .
Hence by Perron-Frobenius Theorem,
ρ(Γ) = ρ(B) ≤ max {ri } ≤ ϕℓ .
1≤i≤n
The
first part of this theorem follows.
.
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Spectral radius and average 2-degree sequence of a graph
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.
Proof. (equality part)
.
To prove the sufficient condition, suppose M1 = M2 = · · · = Mn . Then
ρ(Γ) = M1 = ϕℓ . Hence the equality follows.
To prove the necessary condition, suppose ρ(Γ) = ϕℓ . Hence
r1 = r2 = · · · = rn = ϕℓ , and the equalities related hold. In particular,
b = aik
dk
di
for any 1 ≤ i ≤ n and 1 ≤ k ≤ ℓ − 1 with k ̸= i and xk − 1 > 0, and
Mℓ = Mn . We separate the condition into three cases:
1. M = M : Clearly M = M .
1
1
n
ℓ
. Mt−1 > Mt = Mℓ for some 3 ≤ t ≤ ℓ: Then xk > 1 for 1 ≤ k ≤ t − 1.
Hence b = a12 dd12 = a21 dd12 = 1, and di = n − 1 for all i = 1, 2, · · · , n.
3. M > M = M : Then x > 1. Hence b = a d /d for 2 ≤ i ≤ n.
1
2
1
i1 1 i
ℓ
2
.
Hence d1 = n − 1 and d2 = d3 = · · · = dn = (n − 1)/b. Then
(n − 1)/b = M1 > M2 = Mn = (n − 1)/b − 1 + b.
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Spectral radius and average 2-degree sequence of a graph
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Comparing two theorems
.
Theorem (Chen, Pan and Zhang, 2011)
.
Let a := max {di /dj | 1 ≤ i, j ≤ n}. Then
√
M2 − a + (M2 + a)2 + 4a(M1 − M2 )
ρ(Γ) ≤
,
2
with
equality if and only if Γ is pseudo-regular.
.
.
Theorem
.
For any b ≥ max {di /dj | i ∼ j} and 1 ≤ ℓ ≤ n,
√
∑
Mℓ − b + (Mℓ + b)2 + 4b ℓ−1
i=1 (Mi − Mℓ )
ρ(Γ) ≤
,
2
with
equality if and only if Γ is pseudo-regular.
.
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Spectral radius and average 2-degree sequence of a graph
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Comparing two theorems
Given a decreasing sequence M1 ≥ M2 ≥ · · · ≥ Mn of positive integers,
consider the functions
√
∑
Mℓ − x + (Mℓ + x)2 + 4x ℓ−1
i=1 (Mi − Mℓ )
ϕℓ (x) =
2
for x ∈ [1, ∞).
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Spectral radius and average 2-degree sequence of a graph
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Comparing two theorems
Given a decreasing sequence M1 ≥ M2 ≥ · · · ≥ Mn of positive integers,
consider the functions
√
∑
Mℓ − x + (Mℓ + x)2 + 4x ℓ−1
i=1 (Mi − Mℓ )
ϕℓ (x) =
2
for x ∈ [1, ∞).
.
Proposition
.
.For any 1 ≤ ℓ ≤ n, ϕℓ (x) is increasing on [1, ∞).
黃喻培 (Dep. of A. Math., NCTU)
Spectral radius and average 2-degree sequence of a graph
14 / 18
Comparing two theorems
Given a decreasing sequence M1 ≥ M2 ≥ · · · ≥ Mn of positive integers,
consider the functions
√
∑
Mℓ − x + (Mℓ + x)2 + 4x ℓ−1
i=1 (Mi − Mℓ )
ϕℓ (x) =
2
for x ∈ [1, ∞).
.
Proposition
.
.For any 1 ≤ ℓ ≤ n, ϕℓ (x) is increasing on [1, ∞).
.
Proposition
.
Suppose Ms > Ms+1 for some 1 ≤ s ≤ n − 1, and let the symbol ≽ denote
> or =. Then
ϕs (x) ≽ ϕs+1 (x) iff
.
s
∑
Mi ≽ xs(s − 1).
i=1
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Spectral radius and average 2-degree sequence of a graph
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Comparing two theorems
.
Theorem (Chen, Pan and Zhang, 2011)
.
Let a := max {di /dj | 1 ≤ i, j ≤ n}. Then
√
M2 − a + (M2 + a)2 + 4a(M1 − M2 )
ρ(Γ) ≤
,
2
with
equality if and only if Γ is pseudo-regular.
.
.
Theorem
.
For any b ≥ max {di /dj | i ∼ j} and 1 ≤ ℓ ≤ n,
√
∑
Mℓ − b + (Mℓ + b)2 + 4b ℓ−1
i=1 (Mi − Mℓ )
ρ(Γ) ≤
,
2
with
equality if and only if Γ is pseudo-regular.
.
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Spectral radius and average 2-degree sequence of a graph
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A graph with ϕ2 > ϕ3
.
Example
.
In the following graph, M1 =√M2 = 4, M3 = 7/2, b = 4/3, ϕ1 = ϕ2 = 4,
ϕ
. 3 + 3.762 and ρ(Γ) = 1 + 7 + 3.646.
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Spectral radius and average 2-degree sequence of a graph
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Another graph with ϕ2 > ϕ3
.
Example
.
In the following graph, M1 = 14/3, M2 = 4, M3 = 3.5, a = 5, b = 4,
.ϕ1 = 14/3, ϕ2 (a) + 4.356, ϕ2 (b) + 4.320, ϕ3 + 4.077, and ρ(Γ) + 3.500.
4
3.5
3.5
3.2
3.2
3.5
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3.5
14/3
3.5
Spectral radius and average 2-degree sequence of a graph
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Thank you for your attention!
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Spectral radius and average 2-degree sequence of a graph
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