For a finite noncyclic group
, let Cyc(
) be the set of elements
of
such that 〈
,
〉 is cyclic for each
of
. The noncyclic graph of
is a graph with the vertex set
∖ Cyc(
), having an edge between two ...distinct vertices
and
if 〈
,
〉 is not cyclic. In this paper, we classify all finite noncyclic groups whose noncyclic graphs are
-free, where
is a star and 3 ≤
≤ 6.
The power graph of a finite group is the graph whose vertex set is the group, two distinct elements being adjacent if one is a power of the other. The enhanced power graph of a finite group is the ...graph whose vertex set consists of all elements of the group, in which two vertices are adjacent if they generate a cyclic subgroup. In this paper, we give a complete description of finite groups with enhanced power graphs admitting a perfect code. In addition, we describe all groups in the following two classes of finite groups: the class of groups with power graphs admitting a total perfect code, and the class of groups with enhanced power graphs admitting a total perfect code. Furthermore, we characterize several families of finite groups with power graphs admitting a perfect code, and several other families of finite groups with power graphs which do not admit perfect codes.
Let G be a finite group. The reduced power graph of G is the undirected graph whose vertex set is G, and two distinct vertices x and y are adjacent if 〈x〉 ⊂ 〈y〉 or 〈y〉 ⊂ 〈x〉. In this paper, we give ...tight upper and lower bounds for the metric dimension of the reduced power graph of a finite group. As applications, we compute the metric dimension of the reduced power graph of a 𝒫-group, a cyclic group, a dihedral group, a generalized quaternion group, and a group of odd order.
Let
G
be a finite group. The intersection graph Δ
G
of
G
is an undirected graph without loops and multiple edges defined as follows: the vertex set is the set of all proper nontrivial subgroups of
G
..., and two distinct vertices
X
and
Y
are adjacent if
X
∩
Y
≠ 1, where 1 denotes the trivial subgroup of order 1. A question was posed by Shen (2010) whether the diameters of intersection graphs of finite non-abelian simple groups have an upper bound. We answer the question and show that the diameters of intersection graphs of finite non-abelian simple groups have an upper bound 28. In particular, the intersection graph of a finite non-abelian simple group is connected.
•Machine learning has great potential for assessing vegetation restoration and greenness.•RFR model using meteorological factors explained 80% of NDVI variation•Climate and human activities affect ...greening in China's ecological engineering zones•Human activity was the most important factor associated with NDVI increase
A series of policies and laws have been implemented to address climate change impacts in China since the 1980s. One of the most notable policies is ecological restoration engineering. However, there are many environmental factors that affect vegetation in the ecological restoration engineering zones. The relationships among different factors cannot be explained well by traditional statistical methods due to the existence of hidden non-linear features. Moreover, it is difficult to adopt threshold methods to accurately define vegetation areas fully, or to quantitatively analyze and assess the effects of climate factors and human activities on vegetation changes. The objective of this study was to determine vegetation area and distribution using Landsat TM/ETM/OLI images combined with a support vector machine (SVM) classification model. We analyzed the dynamic characteristics of vegetation area and greenness (NDVI, Normalized Difference Vegetation Index) in China's ecological restoration engineering zones from 1990 to 2015. Based on random forest regression (RFR) with a residual analysis method, the contributions of meteorological factors and human activities to vegetation greenness changes were quantitatively evaluated. Vegetation area and NDVI changed significantly in the study areas, increasing by more than 50% and 40%, respectively, from 1990 to 2015. Temperature, sunshine hours, and precipitation impacted vegetation greenness, which caused NDVI fluctuations in specific years. However, the NDVI increase was difficult to explain fully with meteorological factors. Using cross-validation, we predicted about 80% of the observed NDVI variation occurring from 1984 to 1994. Nine meteorological factors were related to vegetation growth, of which the average temperature, minimum temperature, maximum temperature, and average relative humidity were most critical. The combined effect of the nine climatic factors contributed less to NDVI increase than human activities. Human activity was the most important factor associated with NDVI increase, with contributions of more than 100% in most study areas. Human activities derived from national or local policies had large impacts on vegetation changes. The methods and results of this study can help to understand vegetation changes observed in ecological zones and provide guidance for evaluating ecological restoration policies.
We characterize the strong metric dimension of the power graph of a finite group. As applications, we compute the strong metric dimension of the power graph of a cyclic group, an abelian group, a ...dihedral group and a generalized quaternion group.
Graphs associated with groups and other algebraic structures have been actively investigated, since they have valuable applications in data mining. For a finite group G, let ΓG be the graph with the ...non-identity elements of G as the vertex set, and two vertices are adjacent if they respectively lie in two conjugate proper subgroups of G. ΓG is called the generalized power graph with respect to G. This paper explores how the graph theoretical properties of ΓG can affect on the group theoretical properties of G.
As many data in practical applications occur or can be arranged in multiview forms, multiview clustering utilizing certain complementary and heterogeneous information in various views to promote the ...clustering performance, has received much attention recently. Among varieties of methods, graph-based unsupervised learning methods are an essential approach for learning intrinsic structure relations of multiview data for clustering. Most of them firstly integrate information from each view into a consensus graph, which is then fed into the classic spectral clustering to achieve clustering. Such a two-step clustering paradigm is difficult to obtain the optimal clustering results even though every step performs individual optimization. This paper integrates multi-graph construction, consensus graph construction, and clustering in a unified learning framework, which can simultaneously consider the consistency and complementarity of multiview data to provide the clustering results directly. Moreover, we treat each view differently by automatic weight learning. Specifically, multi-graph learning, consensus graph learning, and weight learning are seamlessly integrated so that the related variables can be iteratively updated in the unified optimization framework–the clustering results towards an overall optimum. Comprehensive experiments on real multiview datasets verify the superiority of the proposed method over other state-of-the-art baselines in terms of three clustering evaluation metrics.
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