On (Signless) Laplacian eigenvalues of graphs · On (Signless) Laplacian eigenvalues of graphs...
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On (Signless) Laplacian eigenvalues of graphs
Kinkar Chandra Das
Department of Mathematics,Sungkyunkwan University, Rep. of Korea
Spectra of graphs and applications 2016,in honor of Prof. Dragos Cvetkovic for his 75th birthday,Serbian Academy of Sciences and Arts, Belgrade, Serbia.
May 20, 2016 (Joint work with M. Liu)
( )
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Adjacency and Signless Laplacian Matrix
Conjecture 1 [1,2]:
Let G be a connected graph of order n > 3. Then
q1 − 2λ1 ≤ n− 2√n− 1
with equality holding if and only if G ∼= K1, n−1.
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Adjacency and Signless Laplacian Matrix
Conjecture 1 [1,2]:
Let G be a connected graph of order n > 3. Then
q1 − 2λ1 ≤ n− 2√n− 1
with equality holding if and only if G ∼= K1, n−1.
Conjecture 2 [1,2]:
Let G be a connected graph n > 3,
1−√n − 1 ≤ q2 − λ1 ≤ n − 2−
√2 (n − 2)
with equality holding iff G ∼= K1, n−1 (lower) and G ∼= K2, n−2 (upper).
[1] M. Aouchiche, P. Hansen, A survey of automated conjectures...., Linear Algebra Appl. 432 (2010) 2293-2322.
[2] D.Cvetkovic, P. Rowlinson, S. K. Simic, Eigenvalue bounds for the signless Laplacian, Publ. Inst. Math.
(Beogr ) (N S ) 81 (95) (2007) 11–27
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Relation between q1 and λ1
Theorem [3]:
Let G be a connected graph of order n > 4. Then
q2 − λ1 ≥ 1−√n − 1
with equality holding if and only if G ∼= K1, n−1 or G ∼= K5.
[3] K. C. Das, Proof of conjecture involving the second largest signless Laplacian eigenvalue and the index of
graphs, Linear Algebra Appl. 435 (2011) 2420–2424.
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Adjacency and Signless Laplacian Matrix
Partial Proof of Conjecture 1:
Let x = (x1, x2, . . . , xn)T be an unit eigenvector corresponding eigenvalue
q1 of Q(G ). Then
q1 =∑
vi vj∈E(G)
(xi + xj)2
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Adjacency and Signless Laplacian Matrix
Partial Proof of Conjecture 1:
Let x = (x1, x2, . . . , xn)T be an unit eigenvector corresponding eigenvalue
q1 of Q(G ). Then
q1 =∑
vi vj∈E(G)
(xi + xj)2 =
∑vi vj∈E(G)
(xi − xj)2 + 4
∑vi vj∈E(G)
xixj .
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Adjacency and Signless Laplacian Matrix
Partial Proof of Conjecture 1:
Let x = (x1, x2, . . . , xn)T be an unit eigenvector corresponding eigenvalue
q1 of Q(G ). Then
q1 =∑
vi vj∈E(G)
(xi + xj)2 =
∑vi vj∈E(G)
(xi − xj)2 + 4
∑vi vj∈E(G)
xixj .
Rayleigh-Ritz theorem,
λ1 ≥ 2∑
vi vj∈E(G)
xixj .
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Adjacency and Signless Laplacian Matrix
Partial Proof of Conjecture 1:
Let x = (x1, x2, . . . , xn)T be an unit eigenvector corresponding eigenvalue
q1 of Q(G ). Then
q1 =∑
vi vj∈E(G)
(xi + xj)2 =
∑vi vj∈E(G)
(xi − xj)2 + 4
∑vi vj∈E(G)
xixj .
Rayleigh-Ritz theorem,
λ1 ≥ 2∑
vi vj∈E(G)
xixj .
Hence
q1 − 2λ1 ≤n∑
i=1
di x2i − λ1
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Adjacency and Signless Laplacian Matrix
Partial Proof of Conjecture 1:
Let x = (x1, x2, . . . , xn)T be an unit eigenvector corresponding eigenvalue
q1 of Q(G ). Then
q1 =∑
vi vj∈E(G)
(xi + xj)2 =
∑vi vj∈E(G)
(xi − xj)2 + 4
∑vi vj∈E(G)
xixj .
Rayleigh-Ritz theorem,
λ1 ≥ 2∑
vi vj∈E(G)
xixj .
Hence
q1 − 2λ1 ≤n∑
i=1
di x2i − λ1 ≤ Δ− λ1.
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Adjacency and Signless Laplacian Matrix
Partial Proof of Conjecture 1:
If Δ ≤ n − 2√n − 1, then the Conjecture 1 holds.
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Adjacency and Signless Laplacian Matrix
Partial Proof of Conjecture 1:
If Δ ≤ n − 2√n − 1, then the Conjecture 1 holds.Otherwise,
Δ > n − 2√n − 1.
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Adjacency and Signless Laplacian Matrix
Partial Proof of Conjecture 1:
If Δ ≤ n − 2√n − 1, then the Conjecture 1 holds.Otherwise,
Δ > n − 2√n − 1.
If m ≥ n2
[Δ− n + 2
√n − 1
], then also Conjecture 1 holds.
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Adjacency and Signless Laplacian Matrix
Partial Proof of Conjecture 1:
If Δ ≤ n − 2√n − 1, then the Conjecture 1 holds.Otherwise,
Δ > n − 2√n − 1.
If m ≥ n2
[Δ− n + 2
√n − 1
], then also Conjecture 1 holds.
Still we have to prove our Conjecture 1 for Δ > n − 2√n − 1 and
m < n2
[Δ− n+ 2
√n− 1
].
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Adjacency and Signless Laplacian Matrix
Partial Proof of Conjecture 1:
If Δ ≤ n − 2√n − 1, then the Conjecture 1 holds.Otherwise,
Δ > n − 2√n − 1.
If m ≥ n2
[Δ− n + 2
√n − 1
], then also Conjecture 1 holds.
Still we have to prove our Conjecture 1 for Δ > n − 2√n − 1 and
m < n2
[Δ− n+ 2
√n− 1
].
q1 − 2λ1 ≤∑
vi vj∈E(G)
(xi − xj)2
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Adjacency and Signless Laplacian Matrix
Partial Proof of Conjecture 1:
If Δ ≤ n − 2√n − 1, then the Conjecture 1 holds.Otherwise,
Δ > n − 2√n − 1.
If m ≥ n2
[Δ− n + 2
√n − 1
], then also Conjecture 1 holds.
Still we have to prove our Conjecture 1 for Δ > n − 2√n − 1 and
m < n2
[Δ− n+ 2
√n− 1
].
q1 − 2λ1 ≤∑
vi vj∈E(G)
(xi − xj)2 ≤ m (xmax − xmin)
2 ≤ n
2
[Δ− n + 2
√n − 1
]
× (xmax − xmin)2 .
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Conjecture 1 for tree
Proof of Conjecture 1 for tree:
Let mi be the average degree of the adjacent vertices of vertex vi .
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Conjecture 1 for tree
Proof of Conjecture 1 for tree:
Let mi be the average degree of the adjacent vertices of vertex vi .We have
q1 ≤ max1≤i≤n
(di +mi )
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Conjecture 1 for tree
Proof of Conjecture 1 for tree:
Let mi be the average degree of the adjacent vertices of vertex vi .We have
q1 ≤ max1≤i≤n
(di +mi ) = dk +mk , (say).
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Conjecture 1 for tree
Proof of Conjecture 1 for tree:
Let mi be the average degree of the adjacent vertices of vertex vi .We have
q1 ≤ max1≤i≤n
(di +mi ) = dk +mk , (say).
Moreover, for Tree
λ1 ≥ max1≤i≤n
√di +mi − 1
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Conjecture 1 for tree
Proof of Conjecture 1 for tree:
Let mi be the average degree of the adjacent vertices of vertex vi .We have
q1 ≤ max1≤i≤n
(di +mi ) = dk +mk , (say).
Moreover, for Tree
λ1 ≥ max1≤i≤n
√di +mi − 1 ≥
√dk +mk − 1.
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Conjecture 1 for tree
Proof of Conjecture 1 for tree:
Let mi be the average degree of the adjacent vertices of vertex vi .We have
q1 ≤ max1≤i≤n
(di +mi ) = dk +mk , (say).
Moreover, for Tree
λ1 ≥ max1≤i≤n
√di +mi − 1 ≥
√dk +mk − 1.
Then
q1 − 2λ1 ≤ (√
dk +mk − 1− 1)2
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Conjecture 1 for tree
Proof of Conjecture 1 for tree:
Let mi be the average degree of the adjacent vertices of vertex vi .We have
q1 ≤ max1≤i≤n
(di +mi ) = dk +mk , (say).
Moreover, for Tree
λ1 ≥ max1≤i≤n
√di +mi − 1 ≥
√dk +mk − 1.
Then
q1 − 2λ1 ≤ (√
dk +mk − 1− 1)2 ≤ (√n− 1− 1)2 = n − 2
√n − 1.
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Algebraic Connectivity
Conjecture 3 [4,5,6]:
a(G )/δ(G ) is minimum for graph composed of 2 triangles linked with apath.
[4] M. Aouchiche, Comparaison Automatisee d’Invariants en Theorie des Graphes, Ph.D. Thesis, Ecole
Polytechnique de Montreal, February 2006.
[5] M. Aouchiche, G. Caporossi, P. Hansen, Variable neighborhood search for extremal graphs. 20. Automated
comparison of graph invariants, MATCH Commun. Math. Comput. Chem. 58 (2007) 365–384.
[6] M. Aouchiche, P. Hansen, A survey of automated conjectures in spectral graph theory, Linear Algebra Appl. 432
(2010) 2293–2322.
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Notation:
A path with n vertices is denoted by Pn.
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Notation:
A path with n vertices is denoted by Pn.
The near path Qn is the tree on n vertices obtained from a path Pn−1 :v1v2 · · · vn−2vn−1 by attaching a new pendant edge vn−2vn at vn−2.
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Notation:
A path with n vertices is denoted by Pn.
The near path Qn is the tree on n vertices obtained from a path Pn−1 :v1v2 · · · vn−2vn−1 by attaching a new pendant edge vn−2vn at vn−2.
Let Wn be the tree on n vertices obtained from a path Pn−2 :v2v3 · · · vn−2vn−1 by attaching a new pendant edge vn−2vn at vn−2 andanother new pendant edge v1v3 at v3, respectively.
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Notation:
A path with n vertices is denoted by Pn.
The near path Qn is the tree on n vertices obtained from a path Pn−1 :v1v2 · · · vn−2vn−1 by attaching a new pendant edge vn−2vn at vn−2.
Let Wn be the tree on n vertices obtained from a path Pn−2 :v2v3 · · · vn−2vn−1 by attaching a new pendant edge vn−2vn at vn−2 andanother new pendant edge v1v3 at v3, respectively.
Let Q ′n = Qn + vn−1vn, W
′n = Wn + v1v2 and W ′′
n = Wn + v1v2 + vn−1vn.
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Notation:
A path with n vertices is denoted by Pn.
The near path Qn is the tree on n vertices obtained from a path Pn−1 :v1v2 · · · vn−2vn−1 by attaching a new pendant edge vn−2vn at vn−2.
Let Wn be the tree on n vertices obtained from a path Pn−2 :v2v3 · · · vn−2vn−1 by attaching a new pendant edge vn−2vn at vn−2 andanother new pendant edge v1v3 at v3, respectively.
Let Q ′n = Qn + vn−1vn, W
′n = Wn + v1v2 and W ′′
n = Wn + v1v2 + vn−1vn.
Let Zn be the tree on n vertices obtained from a path Pn−1 :v1v2 · · · vn−2vn−1 by attaching a new pendant edge vn−3vn at vn−3.
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Inequality
Lemma 1:
For x > 0,
x > sin x > x − x3
6.
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Inequality
Lemma 1:
For x > 0,
x > sin x > x − x3
6.
Lemma 2:
For any positive integer n > 3,
sin( π
2 n
)>
1√2sin
(π
2 (n − 1)
).
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Inequality
Proof:
For x > 0,
sin( π
2n
)>
π
2n− π3
48 n3>
π
2√2 (n − 1)
>1√2sin
(π
2 (n − 1)
).
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Laplacian Eigenvalues
Lemma 3:
The Laplacian eigenvalues of path Pn are
2 + 2 cos(π i
n
), i = 1, 2, . . . , n − 1 and 0.
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Algebraic Connectivity
Definition:
We define
Ad(G ) =a(G )
δ(G ).
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Algebraic Connectivity
Definition:
We define
Ad(G ) =a(G )
δ(G ).
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Algebraic Connectivity
Definition:
We define
Ad(G ) =a(G )
δ(G ).
Conjecture 3:
Let G be a connected graph of order n > 3 and minimum degree δ. Then
Ad(G ) ≥ Ad(W ′′n ) (1)
with equality holding if and only if G ∼= W ′′n .
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Algebraic Connectivity
Theorem 1:
Let G be a connected graph of order n > 3 and minimum degree δ. If δ ≤ 2or δ ≥ n/2, then
Ad(G ) ≥ Ad(W ′′n ) (2)
with equality holding if and only if G ∼= W ′′n .
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Algebraic Connectivity
Theorem 1:
Let G be a connected graph of order n > 3 and minimum degree δ. If δ ≤ 2or δ ≥ n/2, then
Ad(G ) ≥ Ad(W ′′n ) (2)
with equality holding if and only if G ∼= W ′′n .
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Algebraic Connectivity
Theorem 1:
Let G be a connected graph of order n > 3 and minimum degree δ. If δ ≤ 2or δ ≥ n/2, then
Ad(G ) ≥ Ad(W ′′n ) (2)
with equality holding if and only if G ∼= W ′′n .
Proof:
For 4 ≤ n ≤ 8, one can easily check the result by Sage. Otherwise, n ≥ 9.
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Lemma 4 [7]:
Let G be a connected graph of order n ≥ 9 andG /∈ {Pn, Qn, Q
′n, Wn, W
′n, W
′′n }. Then we have
a(Pn) < a(Qn) = a(Q ′n) < a(Wn) = a(W ′
n) = a(W ′′n ) < a(G ), a(Wn) < a(Zn
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Lemma 4 [7]:
Let G be a connected graph of order n ≥ 9 andG /∈ {Pn, Qn, Q
′n, Wn, W
′n, W
′′n }. Then we have
a(Pn) < a(Qn) = a(Q ′n) < a(Wn) = a(W ′
n) = a(W ′′n ) < a(G ), a(Wn) < a(Zn
Lemma 5 [8]:
If v is a pendent vertex, then a(G ) ≤ a(G − v).
[7] J.-Y. Shao, J.-M. Guo, H.-Y. Shan, The ordering of trees and connected graphs by algebraic connectivity, Linear
Algebra Appl. 428 (2008) 1421–1438.
[8] J. X. Li, J.-M. Guo, W. C. Shiu, The Smallest Values of Algebraic Connectivity for Trees, Acta Mathematica
Sinica, English Series 28 (10) (2012) 2021–2032.
![Page 41: On (Signless) Laplacian eigenvalues of graphs · On (Signless) Laplacian eigenvalues of graphs Kinkar Chandra Das Department of Mathematics, Sungkyunkwan University, Rep. of Korea](https://reader035.fdocuments.net/reader035/viewer/2022071510/612dd34f1ecc515869426e14/html5/thumbnails/41.jpg)
Proof:
Case (i): δ = 1.
If G /∈ {Pn, Qn, Q′n, Wn, W
′n}, then by Lemma 4, we get
Ad(G ) = a(G )
![Page 42: On (Signless) Laplacian eigenvalues of graphs · On (Signless) Laplacian eigenvalues of graphs Kinkar Chandra Das Department of Mathematics, Sungkyunkwan University, Rep. of Korea](https://reader035.fdocuments.net/reader035/viewer/2022071510/612dd34f1ecc515869426e14/html5/thumbnails/42.jpg)
Proof:
Case (i): δ = 1.
If G /∈ {Pn, Qn, Q′n, Wn, W
′n}, then by Lemma 4, we get
Ad(G ) = a(G ) > a(W ′′n )
![Page 43: On (Signless) Laplacian eigenvalues of graphs · On (Signless) Laplacian eigenvalues of graphs Kinkar Chandra Das Department of Mathematics, Sungkyunkwan University, Rep. of Korea](https://reader035.fdocuments.net/reader035/viewer/2022071510/612dd34f1ecc515869426e14/html5/thumbnails/43.jpg)
Proof:
Case (i): δ = 1.
If G /∈ {Pn, Qn, Q′n, Wn, W
′n}, then by Lemma 4, we get
Ad(G ) = a(G ) > a(W ′′n ) >
a(W ′′n )
2
![Page 44: On (Signless) Laplacian eigenvalues of graphs · On (Signless) Laplacian eigenvalues of graphs Kinkar Chandra Das Department of Mathematics, Sungkyunkwan University, Rep. of Korea](https://reader035.fdocuments.net/reader035/viewer/2022071510/612dd34f1ecc515869426e14/html5/thumbnails/44.jpg)
Proof:
Case (i): δ = 1.
If G /∈ {Pn, Qn, Q′n, Wn, W
′n}, then by Lemma 4, we get
Ad(G ) = a(G ) > a(W ′′n ) >
a(W ′′n )
2= Ad(a(W ′′
n )).
![Page 45: On (Signless) Laplacian eigenvalues of graphs · On (Signless) Laplacian eigenvalues of graphs Kinkar Chandra Das Department of Mathematics, Sungkyunkwan University, Rep. of Korea](https://reader035.fdocuments.net/reader035/viewer/2022071510/612dd34f1ecc515869426e14/html5/thumbnails/45.jpg)
Proof:
Case (i): δ = 1.
If G /∈ {Pn, Qn, Q′n, Wn, W
′n}, then by Lemma 4, we get
Ad(G ) = a(G ) > a(W ′′n ) >
a(W ′′n )
2= Ad(a(W ′′
n )).
Moreover, we have
Ad(Pn) < Ad(Qn)
![Page 46: On (Signless) Laplacian eigenvalues of graphs · On (Signless) Laplacian eigenvalues of graphs Kinkar Chandra Das Department of Mathematics, Sungkyunkwan University, Rep. of Korea](https://reader035.fdocuments.net/reader035/viewer/2022071510/612dd34f1ecc515869426e14/html5/thumbnails/46.jpg)
Proof:
Case (i): δ = 1.
If G /∈ {Pn, Qn, Q′n, Wn, W
′n}, then by Lemma 4, we get
Ad(G ) = a(G ) > a(W ′′n ) >
a(W ′′n )
2= Ad(a(W ′′
n )).
Moreover, we have
Ad(Pn) < Ad(Qn) = Ad(Q ′n)
![Page 47: On (Signless) Laplacian eigenvalues of graphs · On (Signless) Laplacian eigenvalues of graphs Kinkar Chandra Das Department of Mathematics, Sungkyunkwan University, Rep. of Korea](https://reader035.fdocuments.net/reader035/viewer/2022071510/612dd34f1ecc515869426e14/html5/thumbnails/47.jpg)
Proof:
Case (i): δ = 1.
If G /∈ {Pn, Qn, Q′n, Wn, W
′n}, then by Lemma 4, we get
Ad(G ) = a(G ) > a(W ′′n ) >
a(W ′′n )
2= Ad(a(W ′′
n )).
Moreover, we have
Ad(Pn) < Ad(Qn) = Ad(Q ′n) < Ad(Wn)
![Page 48: On (Signless) Laplacian eigenvalues of graphs · On (Signless) Laplacian eigenvalues of graphs Kinkar Chandra Das Department of Mathematics, Sungkyunkwan University, Rep. of Korea](https://reader035.fdocuments.net/reader035/viewer/2022071510/612dd34f1ecc515869426e14/html5/thumbnails/48.jpg)
Proof:
Case (i): δ = 1.
If G /∈ {Pn, Qn, Q′n, Wn, W
′n}, then by Lemma 4, we get
Ad(G ) = a(G ) > a(W ′′n ) >
a(W ′′n )
2= Ad(a(W ′′
n )).
Moreover, we have
Ad(Pn) < Ad(Qn) = Ad(Q ′n) < Ad(Wn) = Ad(W ′
n).
![Page 49: On (Signless) Laplacian eigenvalues of graphs · On (Signless) Laplacian eigenvalues of graphs Kinkar Chandra Das Department of Mathematics, Sungkyunkwan University, Rep. of Korea](https://reader035.fdocuments.net/reader035/viewer/2022071510/612dd34f1ecc515869426e14/html5/thumbnails/49.jpg)
Proof:
Case (i): δ = 1.
If G /∈ {Pn, Qn, Q′n, Wn, W
′n}, then by Lemma 4, we get
Ad(G ) = a(G ) > a(W ′′n ) >
a(W ′′n )
2= Ad(a(W ′′
n )).
Moreover, we have
Ad(Pn) < Ad(Qn) = Ad(Q ′n) < Ad(Wn) = Ad(W ′
n).
Now we have to show that
Ad(Pn) > Ad(W ′′n )
![Page 50: On (Signless) Laplacian eigenvalues of graphs · On (Signless) Laplacian eigenvalues of graphs Kinkar Chandra Das Department of Mathematics, Sungkyunkwan University, Rep. of Korea](https://reader035.fdocuments.net/reader035/viewer/2022071510/612dd34f1ecc515869426e14/html5/thumbnails/50.jpg)
Proof:
Case (i): δ = 1.
If G /∈ {Pn, Qn, Q′n, Wn, W
′n}, then by Lemma 4, we get
Ad(G ) = a(G ) > a(W ′′n ) >
a(W ′′n )
2= Ad(a(W ′′
n )).
Moreover, we have
Ad(Pn) < Ad(Qn) = Ad(Q ′n) < Ad(Wn) = Ad(W ′
n).
Now we have to show that
Ad(Pn) > Ad(W ′′n ), i.e., a(Wn) = a(W ′′
n )
![Page 51: On (Signless) Laplacian eigenvalues of graphs · On (Signless) Laplacian eigenvalues of graphs Kinkar Chandra Das Department of Mathematics, Sungkyunkwan University, Rep. of Korea](https://reader035.fdocuments.net/reader035/viewer/2022071510/612dd34f1ecc515869426e14/html5/thumbnails/51.jpg)
Proof:
Case (i): δ = 1.
If G /∈ {Pn, Qn, Q′n, Wn, W
′n}, then by Lemma 4, we get
Ad(G ) = a(G ) > a(W ′′n ) >
a(W ′′n )
2= Ad(a(W ′′
n )).
Moreover, we have
Ad(Pn) < Ad(Qn) = Ad(Q ′n) < Ad(Wn) = Ad(W ′
n).
Now we have to show that
Ad(Pn) > Ad(W ′′n ), i.e., a(Wn) = a(W ′′
n ) < 2 a(Pn)
![Page 52: On (Signless) Laplacian eigenvalues of graphs · On (Signless) Laplacian eigenvalues of graphs Kinkar Chandra Das Department of Mathematics, Sungkyunkwan University, Rep. of Korea](https://reader035.fdocuments.net/reader035/viewer/2022071510/612dd34f1ecc515869426e14/html5/thumbnails/52.jpg)
Proof:
Case (i): δ = 1.
If G /∈ {Pn, Qn, Q′n, Wn, W
′n}, then by Lemma 4, we get
Ad(G ) = a(G ) > a(W ′′n ) >
a(W ′′n )
2= Ad(a(W ′′
n )).
Moreover, we have
Ad(Pn) < Ad(Qn) = Ad(Q ′n) < Ad(Wn) = Ad(W ′
n).
Now we have to show that
Ad(Pn) > Ad(W ′′n ), i.e., a(Wn) = a(W ′′
n ) < 2 a(Pn) = 2(2− 2 cos
(πn
))
![Page 53: On (Signless) Laplacian eigenvalues of graphs · On (Signless) Laplacian eigenvalues of graphs Kinkar Chandra Das Department of Mathematics, Sungkyunkwan University, Rep. of Korea](https://reader035.fdocuments.net/reader035/viewer/2022071510/612dd34f1ecc515869426e14/html5/thumbnails/53.jpg)
Proof:
Case (i): δ = 1.
By Lemma 1.2, we have
2 sin2( π
2n
)> sin2
(π
2(n − 1)
)
![Page 54: On (Signless) Laplacian eigenvalues of graphs · On (Signless) Laplacian eigenvalues of graphs Kinkar Chandra Das Department of Mathematics, Sungkyunkwan University, Rep. of Korea](https://reader035.fdocuments.net/reader035/viewer/2022071510/612dd34f1ecc515869426e14/html5/thumbnails/54.jpg)
Proof:
Case (i): δ = 1.
By Lemma 1.2, we have
2 sin2( π
2n
)> sin2
(π
2(n − 1)
), i.e., 1− 2 cos
(πn
)+ cos
(π
n − 1
)> 0
![Page 55: On (Signless) Laplacian eigenvalues of graphs · On (Signless) Laplacian eigenvalues of graphs Kinkar Chandra Das Department of Mathematics, Sungkyunkwan University, Rep. of Korea](https://reader035.fdocuments.net/reader035/viewer/2022071510/612dd34f1ecc515869426e14/html5/thumbnails/55.jpg)
Proof:
Case (i): δ = 1.
By Lemma 1.2, we have
2 sin2( π
2n
)> sin2
(π
2(n − 1)
), i.e., 1− 2 cos
(πn
)+ cos
(π
n − 1
)> 0
Using the above result with some Lemmas 1.3, 1.4 and 1.5, we have
a(Wn) < a(Zn)
![Page 56: On (Signless) Laplacian eigenvalues of graphs · On (Signless) Laplacian eigenvalues of graphs Kinkar Chandra Das Department of Mathematics, Sungkyunkwan University, Rep. of Korea](https://reader035.fdocuments.net/reader035/viewer/2022071510/612dd34f1ecc515869426e14/html5/thumbnails/56.jpg)
Proof:
Case (i): δ = 1.
By Lemma 1.2, we have
2 sin2( π
2n
)> sin2
(π
2(n − 1)
), i.e., 1− 2 cos
(πn
)+ cos
(π
n − 1
)> 0
Using the above result with some Lemmas 1.3, 1.4 and 1.5, we have
a(Wn) < a(Zn) ≤ a(Pn−1)
![Page 57: On (Signless) Laplacian eigenvalues of graphs · On (Signless) Laplacian eigenvalues of graphs Kinkar Chandra Das Department of Mathematics, Sungkyunkwan University, Rep. of Korea](https://reader035.fdocuments.net/reader035/viewer/2022071510/612dd34f1ecc515869426e14/html5/thumbnails/57.jpg)
Proof:
Case (i): δ = 1.
By Lemma 1.2, we have
2 sin2( π
2n
)> sin2
(π
2(n − 1)
), i.e., 1− 2 cos
(πn
)+ cos
(π
n − 1
)> 0
Using the above result with some Lemmas 1.3, 1.4 and 1.5, we have
a(Wn) < a(Zn) ≤ a(Pn−1) = 2− 2 cos
(π
n − 1
)
![Page 58: On (Signless) Laplacian eigenvalues of graphs · On (Signless) Laplacian eigenvalues of graphs Kinkar Chandra Das Department of Mathematics, Sungkyunkwan University, Rep. of Korea](https://reader035.fdocuments.net/reader035/viewer/2022071510/612dd34f1ecc515869426e14/html5/thumbnails/58.jpg)
Proof:
Case (i): δ = 1.
By Lemma 1.2, we have
2 sin2( π
2n
)> sin2
(π
2(n − 1)
), i.e., 1− 2 cos
(πn
)+ cos
(π
n − 1
)> 0
Using the above result with some Lemmas 1.3, 1.4 and 1.5, we have
a(Wn) < a(Zn) ≤ a(Pn−1) = 2− 2 cos
(π
n − 1
)< 2
(2− 2 cos
(πn
))
![Page 59: On (Signless) Laplacian eigenvalues of graphs · On (Signless) Laplacian eigenvalues of graphs Kinkar Chandra Das Department of Mathematics, Sungkyunkwan University, Rep. of Korea](https://reader035.fdocuments.net/reader035/viewer/2022071510/612dd34f1ecc515869426e14/html5/thumbnails/59.jpg)
Proof:
Case (i): δ = 1.
By Lemma 1.2, we have
2 sin2( π
2n
)> sin2
(π
2(n − 1)
), i.e., 1− 2 cos
(πn
)+ cos
(π
n − 1
)> 0
Using the above result with some Lemmas 1.3, 1.4 and 1.5, we have
a(Wn) < a(Zn) ≤ a(Pn−1) = 2− 2 cos
(π
n − 1
)< 2
(2− 2 cos
(πn
))
= 2 a(Pn).
![Page 60: On (Signless) Laplacian eigenvalues of graphs · On (Signless) Laplacian eigenvalues of graphs Kinkar Chandra Das Department of Mathematics, Sungkyunkwan University, Rep. of Korea](https://reader035.fdocuments.net/reader035/viewer/2022071510/612dd34f1ecc515869426e14/html5/thumbnails/60.jpg)
Proof:
Case (ii): δ = 2.
Then by Lemma 1.4,
Ad(G ) =a(G )
2≥ a(W ′′
n )
2= Ad(W ′′
n )
with equality holding if and only if G ∼= W ′′n .
![Page 61: On (Signless) Laplacian eigenvalues of graphs · On (Signless) Laplacian eigenvalues of graphs Kinkar Chandra Das Department of Mathematics, Sungkyunkwan University, Rep. of Korea](https://reader035.fdocuments.net/reader035/viewer/2022071510/612dd34f1ecc515869426e14/html5/thumbnails/61.jpg)
Proof:
Case (iii): δ ≥ n/2.
By Case (i) and Lemma 1.1, we have
Ad(W ′′n ) =
a(W ′′n )
2
![Page 62: On (Signless) Laplacian eigenvalues of graphs · On (Signless) Laplacian eigenvalues of graphs Kinkar Chandra Das Department of Mathematics, Sungkyunkwan University, Rep. of Korea](https://reader035.fdocuments.net/reader035/viewer/2022071510/612dd34f1ecc515869426e14/html5/thumbnails/62.jpg)
Proof:
Case (iii): δ ≥ n/2.
By Case (i) and Lemma 1.1, we have
Ad(W ′′n ) =
a(W ′′n )
2< 2− 2 cos
(πn
)
![Page 63: On (Signless) Laplacian eigenvalues of graphs · On (Signless) Laplacian eigenvalues of graphs Kinkar Chandra Das Department of Mathematics, Sungkyunkwan University, Rep. of Korea](https://reader035.fdocuments.net/reader035/viewer/2022071510/612dd34f1ecc515869426e14/html5/thumbnails/63.jpg)
Proof:
Case (iii): δ ≥ n/2.
By Case (i) and Lemma 1.1, we have
Ad(W ′′n ) =
a(W ′′n )
2< 2− 2 cos
(πn
)= 4 sin2
( π
2n
)
![Page 64: On (Signless) Laplacian eigenvalues of graphs · On (Signless) Laplacian eigenvalues of graphs Kinkar Chandra Das Department of Mathematics, Sungkyunkwan University, Rep. of Korea](https://reader035.fdocuments.net/reader035/viewer/2022071510/612dd34f1ecc515869426e14/html5/thumbnails/64.jpg)
Proof:
Case (iii): δ ≥ n/2.
By Case (i) and Lemma 1.1, we have
Ad(W ′′n ) =
a(W ′′n )
2< 2− 2 cos
(πn
)= 4 sin2
( π
2n
)<
π2
n2.
![Page 65: On (Signless) Laplacian eigenvalues of graphs · On (Signless) Laplacian eigenvalues of graphs Kinkar Chandra Das Department of Mathematics, Sungkyunkwan University, Rep. of Korea](https://reader035.fdocuments.net/reader035/viewer/2022071510/612dd34f1ecc515869426e14/html5/thumbnails/65.jpg)
Proof:
Case (iii): δ ≥ n/2.
By Case (i) and Lemma 1.1, we have
Ad(W ′′n ) =
a(W ′′n )
2< 2− 2 cos
(πn
)= 4 sin2
( π
2n
)<
π2
n2.
Since n ≥ 5, one can easily see that
n
(1− π2
n2
)> n − 2 .
![Page 66: On (Signless) Laplacian eigenvalues of graphs · On (Signless) Laplacian eigenvalues of graphs Kinkar Chandra Das Department of Mathematics, Sungkyunkwan University, Rep. of Korea](https://reader035.fdocuments.net/reader035/viewer/2022071510/612dd34f1ecc515869426e14/html5/thumbnails/66.jpg)
Proof:
Case (iii): δ ≥ n/2.
By Case (i) and Lemma 1.1, we have
Ad(W ′′n ) =
a(W ′′n )
2< 2− 2 cos
(πn
)= 4 sin2
( π
2n
)<
π2
n2.
Since n ≥ 5, one can easily see that
n
(1− π2
n2
)> n − 2 .
Thus we have
δ
(1− π2
n2
)≥ n
2
(1− π2
n2
)
![Page 67: On (Signless) Laplacian eigenvalues of graphs · On (Signless) Laplacian eigenvalues of graphs Kinkar Chandra Das Department of Mathematics, Sungkyunkwan University, Rep. of Korea](https://reader035.fdocuments.net/reader035/viewer/2022071510/612dd34f1ecc515869426e14/html5/thumbnails/67.jpg)
Proof:
Case (iii): δ ≥ n/2.
By Case (i) and Lemma 1.1, we have
Ad(W ′′n ) =
a(W ′′n )
2< 2− 2 cos
(πn
)= 4 sin2
( π
2n
)<
π2
n2.
Since n ≥ 5, one can easily see that
n
(1− π2
n2
)> n − 2 .
Thus we have
δ
(1− π2
n2
)≥ n
2
(1− π2
n2
)>
n − 2
2
![Page 68: On (Signless) Laplacian eigenvalues of graphs · On (Signless) Laplacian eigenvalues of graphs Kinkar Chandra Das Department of Mathematics, Sungkyunkwan University, Rep. of Korea](https://reader035.fdocuments.net/reader035/viewer/2022071510/612dd34f1ecc515869426e14/html5/thumbnails/68.jpg)
Proof:
Case (iii): δ ≥ n/2.
By Case (i) and Lemma 1.1, we have
Ad(W ′′n ) =
a(W ′′n )
2< 2− 2 cos
(πn
)= 4 sin2
( π
2n
)<
π2
n2.
Since n ≥ 5, one can easily see that
n
(1− π2
n2
)> n − 2 .
Thus we have
δ
(1− π2
n2
)≥ n
2
(1− π2
n2
)>
n − 2
2, i.e.,
π2
n2< 1− n − 2
2 δ.
![Page 69: On (Signless) Laplacian eigenvalues of graphs · On (Signless) Laplacian eigenvalues of graphs Kinkar Chandra Das Department of Mathematics, Sungkyunkwan University, Rep. of Korea](https://reader035.fdocuments.net/reader035/viewer/2022071510/612dd34f1ecc515869426e14/html5/thumbnails/69.jpg)
Case (iii): δ ≥ n/2
Lemma 1.6 [9]:
Let G be a connected graph of order n, minimum degree δ(G )and algebraic connectivity a(G ). Then
a(G )− δ(G ) ≥
⎧⎪⎨⎪⎩
−n − 8 +√n2 + 8n − 16
4if n is even
−n − 3
2if n is odd
Moreover, the equality holds if and only if G ∼= Kn/2, n/2\{e}when n is even (e is any edge), and G ∼= K(n−1)/2, (n−1)/2 ∨ K1
when n is odd.
[9] K. C. Das, Proof of conjectures involving algebraic connectivity of graphs, Linear Algebra Appl. 438 (2013)
3291–3302.
![Page 70: On (Signless) Laplacian eigenvalues of graphs · On (Signless) Laplacian eigenvalues of graphs Kinkar Chandra Das Department of Mathematics, Sungkyunkwan University, Rep. of Korea](https://reader035.fdocuments.net/reader035/viewer/2022071510/612dd34f1ecc515869426e14/html5/thumbnails/70.jpg)
Proof:
Case (iii): δ ≥ n/2.
From Lemma 1.6, we have
−n− 2
2≤ −n− 8 +
√n2 + 8n − 16
4≤ −n− 3
2≤ a(G )− δ.
![Page 71: On (Signless) Laplacian eigenvalues of graphs · On (Signless) Laplacian eigenvalues of graphs Kinkar Chandra Das Department of Mathematics, Sungkyunkwan University, Rep. of Korea](https://reader035.fdocuments.net/reader035/viewer/2022071510/612dd34f1ecc515869426e14/html5/thumbnails/71.jpg)
Proof:
Case (iii): δ ≥ n/2.
From Lemma 1.6, we have
−n− 2
2≤ −n− 8 +
√n2 + 8n − 16
4≤ −n− 3
2≤ a(G )− δ.
Therefore
Ad(G ) =a(G )
δ
![Page 72: On (Signless) Laplacian eigenvalues of graphs · On (Signless) Laplacian eigenvalues of graphs Kinkar Chandra Das Department of Mathematics, Sungkyunkwan University, Rep. of Korea](https://reader035.fdocuments.net/reader035/viewer/2022071510/612dd34f1ecc515869426e14/html5/thumbnails/72.jpg)
Proof:
Case (iii): δ ≥ n/2.
From Lemma 1.6, we have
−n− 2
2≤ −n− 8 +
√n2 + 8n − 16
4≤ −n− 3
2≤ a(G )− δ.
Therefore
Ad(G ) =a(G )
δ≥ 1− n − 2
2 δ
![Page 73: On (Signless) Laplacian eigenvalues of graphs · On (Signless) Laplacian eigenvalues of graphs Kinkar Chandra Das Department of Mathematics, Sungkyunkwan University, Rep. of Korea](https://reader035.fdocuments.net/reader035/viewer/2022071510/612dd34f1ecc515869426e14/html5/thumbnails/73.jpg)
Proof:
Case (iii): δ ≥ n/2.
From Lemma 1.6, we have
−n− 2
2≤ −n− 8 +
√n2 + 8n − 16
4≤ −n− 3
2≤ a(G )− δ.
Therefore
Ad(G ) =a(G )
δ≥ 1− n − 2
2 δ>
π2
n2
![Page 74: On (Signless) Laplacian eigenvalues of graphs · On (Signless) Laplacian eigenvalues of graphs Kinkar Chandra Das Department of Mathematics, Sungkyunkwan University, Rep. of Korea](https://reader035.fdocuments.net/reader035/viewer/2022071510/612dd34f1ecc515869426e14/html5/thumbnails/74.jpg)
Proof:
Case (iii): δ ≥ n/2.
From Lemma 1.6, we have
−n− 2
2≤ −n− 8 +
√n2 + 8n − 16
4≤ −n− 3
2≤ a(G )− δ.
Therefore
Ad(G ) =a(G )
δ≥ 1− n − 2
2 δ>
π2
n2> Ad(W ′′
n ).
![Page 75: On (Signless) Laplacian eigenvalues of graphs · On (Signless) Laplacian eigenvalues of graphs Kinkar Chandra Das Department of Mathematics, Sungkyunkwan University, Rep. of Korea](https://reader035.fdocuments.net/reader035/viewer/2022071510/612dd34f1ecc515869426e14/html5/thumbnails/75.jpg)
Proof:
Remark:
For 3 ≤ δ < n/2, Conjecture 3 is still open.
![Page 76: On (Signless) Laplacian eigenvalues of graphs · On (Signless) Laplacian eigenvalues of graphs Kinkar Chandra Das Department of Mathematics, Sungkyunkwan University, Rep. of Korea](https://reader035.fdocuments.net/reader035/viewer/2022071510/612dd34f1ecc515869426e14/html5/thumbnails/76.jpg)
THANK YOU for attention.