The Sombor index was introduced in 2020, and was soon followed by a remarkably large number of studies, both chemical and mathematical. In this review we collect the existing bounds and extremal ...results related to the Sombor index and its variants.
Let
G
be a graph of order
n
and
μ
be an adjacency eigenvalue of
G
with multiplicity
k
≥ 1. A star complement
H
for
μ
in
G
is an induced subgraph of
G
of order
n
−
k
with no eigenvalue
μ
, and the ...subset
X
=
V
(
G
−
H
) is called a star set for
μ
in
G
. The star complement provides a strong link between graph structure and linear algebra. In this paper, the authors characterize the regular graphs with
K
2,2,
s
(
s
≥ 2) as a star complement for all possible eigenvalues, the maximal graphs with
K
2,2,
s
as a star complement for the eigenvalue
μ
= 1, and propose some questions for further research.
Let ℋ be a uniform hypergraph. In this paper, we obtain several bounds for the spectral radius of ℋ in terms of the parameters such as q-average-degrees, diameter, and characterize the corresponding ...extremal hypergraphs. Moreover, we discuss the change for the spectral radius of a uniform hypergraph after deleting a vertex, and give a comparison of our results with some known ones.
With the increasing running speed, the aerodynamic issues of high-speed trains are being raised and impact the running stability and energy efficiency. The optimization design of the head shape is ...significantly important in improving the aerodynamic performance of high-speed trains. Existing aerodynamic optimization methods are limited by the parametric modeling methods of train heads which are unable to accurately and completely parameterize both global shapes and local details. Due to this reason, they cannot optimize both global and local shapes of train heads. In order to tackle this problem, we propose a novel multi-objective aerodynamic optimization method of high-speed train heads based on the partial differential equation (PDE) parametric modeling. With this method, the half of a train head is parameterized with 17 PDE surface patches which describe global shapes and local details and keep the surface smooth. We take the aerodynamic drag and lift as optimization objectives; divide the optimization design process into two stages: global optimization and local optimization; and develop global and local optimization methods, respectively. In the first stage, the non-dominated sorting genetic algorithm (NSGA-II) is adopted to obtain the framework of the train head with an optimized global shape. In the second stage, Latin hypercube sampling (LHS) is applied in the local shape optimization of the PDE surface patches determined by the optimized framework to improve local details. The effectiveness of our proposed method is demonstrated by better aerodynamic performance achieved from the optimization solutions in global and local optimization stages in comparison with the original high-speed train head.
Identifying certain conditions that ensure the Hamiltonicity of graphs is highly important and valuable due to the fact that determining whether a graph is Hamiltonian is an NP-complete problem.For a ...graph
G
with vertex set
V
(
G
) and edge set
E
(
G
), the first Zagreb index (
M
1
) and second Zagreb index (
M
2
) are defined as
M
1
(
G
)
=
∑
v
i
v
j
∈
E
(
G
)
(
d
G
(
v
i
)
+
d
G
(
v
j
)
)
and
M
2
(
G
)
=
∑
v
i
v
j
∈
E
(
G
)
d
G
(
v
i
)
d
G
(
v
j
)
, where
d
G
(
v
i
)
denotes the degree of vertex
v
i
∈
V
(
G
)
. The difference of Zagreb indices (
Δ
M
) of
G
is defined as
Δ
M
(
G
)
=
M
2
(
G
)
-
M
1
(
G
)
.In this paper, we try to look for the relationship between structural graph theory and chemical graph theory. We obtain some sufficient conditions, with regards to
Δ
M
(
G
)
, for graphs to be
k
-hamiltonian, traceable,
k
-edge-hamiltonian,
k
-connected, Hamilton-connected or
k
-path-coverable.
The Turán number of P9∪P7 Fang, Xiaona; You, Lihua
Computational & applied mathematics,
2024/7, Letnik:
43, Številka:
5
Journal Article
Recenzirano
The Turán number of a graph
F
is the maximum number of edges in any graph on
n
vertices containing no
F
as a subgraph. Let
P
ℓ
denote the path on
ℓ
vertices. A linear forest, denoted by
∪
i
=
1
k
P
ℓ
...i
, is a forest whose connected components are paths. Lidický et al. (Electron J Combin 20(2):62, 2013) considered the Turán number of linear forests for sufficiently large
n
. Yuan and Zhang (J Graph Theory 98(3):499–524, 2021) determined the Turán numbers of the linear forests containing at most one odd path for all
n
and proposed a conjecture, which is confirmed to be true for
P
3
∪
P
2
ℓ
+
1
,
2
P
5
,
3
P
5
,
2
P
7
,
2
P
9
,
3
P
7
and
2
P
3
∪
P
2
ℓ
+
1
. Motivated by these results, we determine the Turán numbers of
P
9
∪
P
7
for all
n
≥
16
and characterize all extremal graphs, which partially confirms the conjecture proposed by Yuan and Zhang.
•Sedimentary rate was ∼60 cm/y based on Fe and Mn contents and hydrological data.•The Mn/Fe ratio cannot reconstruct redox statuses in sediment-rich reservoirs.•Fluvial sediment was the main source ...of sediment Fe and Mn in the studied reservoir.
Water hypoxia intensification in lakes and reservoirs has become a global problem under global warming. The Mn/Fe ratio is frequently employed to reconstruct historical redox conditions, but interpretation of this ratio may be problematic when the accumulation of Fe and Mn is governed by factors in addition to redox processes. We tested the Fe and Mn contents and bacterial diversity of a 250 cm sediment column in a monomictic reservoir. The deposition time frame and sedimentary rate (approximately 60 cm/y) were determined by integrating the Fe and Mn contents with the hydrological time series, providing a sufficient archive to reconstruct the intra-year variation in water qualities. The inapplicability of the Mn/Fe ratio observed because redox processes, mixing patterns, and biological activity jointly affected the net accumulation of Fe and Mn in the sediment. The causes of the vertical variations in the sediment Fe and Mn contents are discussed, considering the runoff sediment input, hydrodynamic and thermodynamic characteristics of this reservoir, and bacterial distributions in the sediment column. High-speed deposition intensively occurred in summer and autumn; thus, during this period, fluvial sediment was the main source of Fe and Mn. Another source was biochemical sedimentation due to mixing and oxygenation in this reservoir, which played a larger role in spring and winter. Our study showed that the easily accessible current hydrological data of the reservoir provide a reference time frame for identifying historical environmental events and that the use of the Fe/Mn ratio alone is inconclusive for interpreting the historic oxygenation regimes of reservoirs. Future applications of this method should consider the individual reservoir characteristics that impact the mobility and net accumulation of Fe and Mn in the sediment.
Let
G
=
(
V
G
,
E
G
)
be a simple connected graph with its vertex set
V
G
and edge set
E
G
. The Mostar index
Mo
(
G
) was defined as
M
o
(
G
)
=
∑
e
=
u
v
∈
E
(
G
)
|
n
u
-
n
v
|
, where
n
u
(resp.,
...n
v
) is the number of vertices whose distance to vertex
u
(resp.,
v
) is smaller than the distance to vertex
v
(resp.,
u
). In this study, we determine the first three minimum Mostar indices of tree-like phenylenes and characterize all the tree-like phenylenes attaining these values. At last, we give some numerical examples and discussion.
In this paper, we obtain some sharp upper and lower bounds for the spectral radii of nonnegative k-uniform tensors (resp., nonnegative tensors) by using the i-th q-times-average slice sum which was ...introduced by C. Lü, L. H. You and Y. F. Huang AIMS Math., 2020, 5(3): 1799–1819. When k = 2, q =1, our results can deduce the main results of D. P. Huang and L. H. You J. Appl. Math., 2016, 3: 1–7. We also find that the upper bounds of nonnegative k-uniform tensors we obtained can not be compared with the upper bounds presented by AIMS Math., 2020, 5(3): 1799–1819 by numerical examples. As applications, we obtain some sharp upper bounds for the α-spectral radii of k-uniform hypergraphs (resp., k-uniform directed hypergraphs).
In this paper, we obtain a sharp upper bound on the spectral radius of a nonnegative
k
-uniform tensor and characterize when this bound is achieved. Furthermore, this result deduces the main result ...in X. Duan and B. Zhou, Sharp bounds on the spectral radius of a nonnegative matrix, Linear Algebra Appl. 439:2961–2970,
2013
for nonnegative matrices; improves the adjacency spectral radius and signless Laplacian spectral radius of a uniform hypergraph for some known results in D.M. Chen, Z.B. Chen and X.D. Zhang, Spectral radius of uniform hypergraphs and degree sequences, Front. Math. China 6:1279–1288,
2017
; and presents some new sharp upper bounds for the adjacency spectral radius and signless Laplacian spectral radius of a uniform directed hypergraph. Moreover, a characterization of a strongly connected
k
-uniform directed hypergraph is obtained.
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