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•The results in this paper expand the results obtained by K. Edwards, M. Horňák, M. Woźniak. The proof technique in this paper is discharging. The important result, that the adjacent-vertex-distinguishing proper edge chromatic number χa′(G)≤Δ(G)+1 for any planar bipartite graph G with maximum degree Δ(G)=9,10, or 11, is obtained in this paper.•We implicitly give the definitions of K2, exceptional...
The purpose of implementing mathematical simulation based on evolution models is to predict the possible risk and to detect potential dangers, reduce the cost of investment, increase revenue and incomes in the Internet of Things (IoT). Investigating topological structure of networks and building up network models take an important part in more specifically to understand some of the networks' more...
To understand the dynamical phenomena of networks, one must know the global properties of networks as well as the local properties such as the degree distribution. So, researching topological structures of complex networks is important and catches many researchers' attention. A class of bound growing network models (BGN-models) has been defined in this article. We determine some properties of the...
Connected Dominating Set Problem is a fundamental problem in connected facility location and studied intensively in computer science and operations research and it is also a central problem in wireless networking. Maximum Leaf Spanning Tree Problem (MLSTP) is related with investigating scale-free networks. We investigate the collapse of graphs since “robust yet fragile” is an important character of...
In bioinformatics research field, researchers from biology, and physics, mathematics, computer science and other disciplines explore the mystery of genetic language by the combination of experimental researches and theoretical analysis, and try reveals the essence of biological genetic information. Bollobás and Riordan pointed out that the definition of “scale-free” in the context of network graph...
Let ƒ be a proper total coloring of G. For each x ∈ V(G), let C(x) denote the set of all colors of the elements incident with or adjacent to x and the color of x. If ∀ u, v ∈ V(G), u ≠ v, we have C(u) ≠ C(v), then ƒ is called a vertex strongly distinguishing total coloring of G. The minimum number k for which there exists a vertex strongly distinguishing total coloring of G using k colors is called...
Graph labelings related with communication networks have been extensively investigated because their applications to real problems. Some results on graph labellings are obtained here. Scale-free graphs are coming from scale-free networks a few years ago. We use spanning trees to study some important characteristics of scale-free networks, and obtain some results related with scale-free networks. Furthermore,...
Results on graph coloring can be used to draw conclusions about scheduling. Graph theory is a sort of models which can be applied in various science fields such as computer science, physics, biology, chemistry, strategy etc. And graph coloring is one of the chief topics in graph research. Suppose G(V,E) is a connect graph with order at least 2, k is a positive integer and ƒ is the mapping from V (G)∪E(G)...
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