Download On the Resistance–Distance Spectral Radius

Graph Theory Conference in honor of Egawa’s 60th birthday, September 10 to 14, 2013
On the Resistance–Distance Spectral Radius
A. Dilek Maden
∗
A. Sinan Cevik
†
In this study, we present our results for the maximum eigenvalue λ1 = λ1 (G)
(i. e., the spectral radius) of the resistance–distance matrix R = (Rij ). We also
provide some lower and upper bounds for λ1 (G) for molecular graphs and a few
Nordhaus–Gaddum–type results .
THEOREM 1. Let G be a connected graph with n ≥ 2 vertices. Then
√ ∑n
√
n
2
∑
R
Rj
i
i=1
≤ λ1 (G) ≤ max
Rij
1≤j≤n
n
Ri
i=1
(1)
where Ri is the sum of i-th row of the matrix R . Moreover equality holds in (1) if
and only if R1 = R2 = · · · = Rn .
COROLLARY 2. Let G be a connected graph with n ≥ 2 vertices. Then
λ1 (G) ≥
with where Kf =
∑
2 Kf
n
(2)
Rij denote the Kirchhoff index. Equality holds if and only if
i<j
R1 = R2 = · · · = Rn or G ∼
= Kn .
Let G be the class of connected graphs whose resistance–distance matrices have
exactly one positive eigenvalue. Denote by S = S(G) the trace of R2 .
THEOREM 3. Let G ∈ G with n ≥ 2 vertices. Then
√
n−1
S(G)
λ1 (G) ≤
n
with equality holding if and only if G ∼
= Kn .
(3)
THEOREM 4. Let G ∈ G with n ≥ 2 . Then
√
S(G)
.
λ1 (G) ≥
2
Equality holds in (4) if and only if G ∼
= K2 .
(4)
∗
Department of Mathematics, Science Faculty, Selcuk University, Campus, Konya, TURKEY.
E-mail:[email protected]
†
E-mail:[email protected]
THEOREM 5. Let G be a connected graph on n > 2 vertices, m edges, maximum
degree ∆ , second maximum degree ∆2 and minimum degree δ. Then
[
]}
{
1
1
(∆2 − δ)2
2
λ1 (G) ≥ 2
+
(n − 2) +
(5)
∆ + 1 2m − ∆ − 1
∆2 δ
and
{
}
1 n
n(n − 3)2
λ1 (G) ≥ 2
+ +
.
(6)
n δ
2m − ∆ − δ − 1
THEOREM 6. Let G be a connected graph on n > 2 vertices and m edges, maximum degree ∆ , second maximum degree ∆2 and minimum degree δ such that its
complement G is also connected. Then
{
[
1
1
1
2
λ1 (G) + λ1 (G) ≥ 2
+
+ (n − 2)
+
∆+1 n−δ
2m − ∆ − 1
] (
)(
)
1
1
(∆2 − δ)2
+
+
+
n(n − 2) − 2m + 1 + δ
2m − ∆ − 1
∆2 δ
)(
)}
(
(∆ − δ2 )2
1
(7)
+
n(n − 2) − 2m + 1 + δ
(n − 1 − δ2 )(n − 1 − ∆)
with equality holding if and only if G ∼
= Kn .
THEOREM 7. Let G be a connected graph on n > 2 vertices and m edges, maximum
degree ∆ and minimum degree δ such that its complement G is also connected. such
that its complement G is also connected. Then
{(
)
1
4
1
λ1 (G) + λ1 (G) ≥ + 2
+
+
n
δ (n − 1 − ∆)
[
]}
1
1
2
+ (n − 3)
+
.
2m − ∆ − δ − 1 n(n − 3) − 2m + 1 + ∆ + δ
THEOREM 8. Let G ∈ G with n > 2 vertices, and let G be connected. Then,
√
]
√
n − 1 [√
λ1 (G) + λ1 (G) ≤
S(G) + n3 + S(G)
n
∼
with equality holding if and only if G = Kn .
References
[ 1 ] R. B. Bapat, Resistance distance in graphs, Math. Student 68 (1999), 87-98.
[ 2 ] K. C. Das, A. D. (Güngör) Maden, A. S. Çevik, On the Kirchoff index and
the resistence distance energy of a graph, MATCH Commun. Math. Comput.
Chem. 67(2) (2012), 541-556.
[ 3 ] D. J. Klein, M. Randić, Resistance distance, J. Math. Chem. 12 (1993), 81-95.
[ 4 ] E. A. Nordhaus, J. W. Gaddum, On complementary graphs, Am. Math. Montly
63 (1956), 175-177.
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