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The coloring theory of graphs is a very important direction in graph theory. The graph coloring problem has a strong application background. Many practical problems such as timetabling problems, frequency allocation problems, traffic arrangements, circuit design, and storage problems can be transformed into graph coloring problems. A substantial amount of scholarly work has been dedicated to the exploration of acyclic coloring, resulting in a plethora of significant findings that significantly contribute to the theoretical framework of vertex coloring in graphs. An acyclic coloring of a graph G refers to a proper vertex coloring where the subgraph resulting from any pair of color classes does not encompass any cycles. We analyze acyclic coloring in splitting graphs and Cartesian products of paths and cycles by using the method of structural coloring, reductive proof, and mathematical induction. Moreover, we present the precise acyclic chromatic numbers for splitting graphs of cycles and Cartesian products of paths and cycles.

The injective chromatic number χ i (G) of a graph G is the minimum number of colors needed to color the vertices of G such that different vertices with the same neighbor are colored differently. In this here, we get some exact values of the injective chromatic number of the edge corona product of some graphs.

For the graph’s vertex coloring, it is required that for every vertex in a graph, the colors used in its open neighborhood or closed neighborhood must be able to form a continuous integer interval. A coloring is called an open neighborhood interval vertex coloring or a closed neighborhood interval vertex coloring of a graph if the neighborhood satisfying the condition is open or closed. In this paper, the interval vertex coloring of cartesian products and strong products of two paths is studied, and the low bound of the interval chromatic number is given.

The coloring theory of graphs is an important part of graph theory research. The key problem of the coloring theory of graphs is to determine the coloring number of each kind of coloring. Traditional coloring concepts mainly include proper vertex coloring, proper edge coloring, proper total coloring, and so on. In recent years, scholars at home and abroad have put forward some new coloring concepts, such as neighbor vertex distinguishing edge (total) coloring, and neighbor sum distinguishing edge (total) coloring, based on traditional coloring concepts and by adding other constraints. Some valuable results have been obtained, which further enrich the theory of graph coloring. For a proper [k] -edge coloring of a graph G, if for any adjacent vertex has a different sum of colors, then the coloring is a neighbor sum distinguishing [k] -edge coloring of G. For a proper [k] -total coloring of a graph G, if for any adjacent vertex has a different sum of colors, then the coloring is a neighbor sum distinguishing [k] -total coloring of G . In this paper, the coloring method and coloring index are determined by the process of induction and deduction and the construction of the dyeing method, and then the rationality of the method is verified by inverse proof and mathematical induction. If G is a simple graph with the order n ≥ 5 , and h n = ( H i ) i ∈{1,2,…,n} is a sequence of disjoint simple graphs, where every H i is a simple graph with the order m ≥ 7 . In this paper, we study the neighbor sum distinguishing edge(total) coloring of the generalized corona product G○h n of G and h n . The results are as follows: (1) If G is a path with order n , h n = ( H i ) i ∈{1,2,…,n} is an alternating sequence of path and cycle with order m . If n is odd, we have χ Σ ′ ( G ∘ h n ) = m + 3 (2) If G is a path with order n , h n = ( H i ) i ∈{1,2,…,n} is an alternating sequence of path and cycle with order m . If n is odd, we have χ Σ ′ ′ ( G ∘ h n ) = m + 4 Due to the late development of neighbor sum distinguishing edge (total) coloring of graphs, the related research results are relatively few. By studying the operation graph of a basic simple graph, we can provide the research basis and reference idea for the corresponding coloring of the general graph class. Therefore, it is of theoretical value to study the neighbor sum distinguishing edge (total) coloring problem of generalized corona products of graphs.

With the wide application of combinatorial optimization problems in optics, we were inspired by optics and studied the interval edge coloring problem of operational graphs. The maximum value of t for an interval t -coloring is denoted by W(G) , which is a proper edge-coloring of a graph G using colors 1, 2, …, t . In this coloring process, all colors are used, and the color of the edges associated with each vertex has a different color, forming an interval of integers. In this paper, we prove that G is interval colorable and we give a lower bound on W(G) if and only if G is a generalized Mycielski graph, double graph or middle graph of the path.

Let G be a simple graph with vertex set V(G) and edge set E(G). A vertex coloring of G is called a star coloring of G if any of the paths of 4 order are bicolored. The minimum number of colors required for a star coloring of G is denoted by χs(G). The corona product of simple graphs G of order m and H of order n is graph G ∘ H with vertex set V(G ∘ H) = {vi|i = 1,2,⋯m}∪{vij|i = 1,2,⋯m, j = 1,2,⋯n}, in which vi is adjacent to every vertex of Hi if and only if, vi ∈ V(G), vij ∈ V(Hi). According to the existing graph dyeing literature, it has become a very important technical means to study the graph dyeing problem by using the graph structure operation. Therefore, it is of great significance to study the star coloring of graphs for studying the acyclic coloring and distance coloring of graphs, the study has strong application background and great theoretical value for computing graphs. In this paper, we find the upper bound of χs(G ∘ H) and the exact values of χs(G ∘ H) of the corona product G ∘ H of two graphs G and H as: χs(G ∘ H) ≤ χs(G) + χs(H); χs(Pm ∘ H) = χs(H) + 2; χs(K1,m ∘ H) = χs(H) + 2; χs(Cn ∘ H) = χs(H) + 2, where n ≠ 5.

The matching energy of a graph G is defined as the sum of the absolute values of thezeros of the matching polynomial of G. The Hosoya index of a graph G is defined as the total number of matchingsof G. In this paper, the matching energy and Hosoya index of the a special class of tricyclic graphs G(m1, m2, m3) were investigated, and orderings of tricyclic graphs G(m1, m2, m3) with respect to matching energy and Hosoya index are obtained.

Due to many of the clustering algorithms based on GAs suffer from degeneracy and are easy to fall in local optima, a novel dynamic genetic algorithm for clustering problems (DGA) is proposed. The algorithm adopted the variable length coding to represent individuals and processed the parallel crossover operation in the subpopulation with individuals of the same length, which allows the DGA algorithm clustering to explore the search space more effectively and can automatically obtain the proper number of clusters and the proper partition from a given data set; the algorithm used the dynamic crossover probability and adaptive mutation probability, which prevented the dynamic clustering algorithm from getting stuck at a local optimal solution. The clustering results in the experiments on three artificial data sets and two real-life data sets show that the DGA algorithm derives better performance and higher accuracy on clustering problems.

The hyperspectral image (HSI) has been widely used in many applications due to its fruitful spectral information. However, the limitation of imaging sensors has reduced its spatial resolution that causes detail loss. One solution is to fuse the low spatial resolution hyperspectral image (LR-HSI) and the panchromatic image (PAN) with inverse features to get the high-resolution hyperspectral image (HR-HSI). Most of the existing fusion methods just focus on small fusion ratios like 4 or 6, which might be impractical for some large ratios' HSI and PAN image pairs. Moreover, the ill-posedness of restoring detail information in HSI with hundreds of bands from PAN image with only one band has not been solved effectively, especially under large fusion ratios. Therefore, a lightweight unmixing-based pan-guided fusion network (Pgnet) is proposed to mitigate this ill-posedness and improve the fusion performance significantly. Note that the fusion process of the proposed network is under the projected low-dimensional abundance subspace with an extremely large fusion ratio of 16. Furthermore, based on the linear and nonlinear relationships between the PAN intensity and abundance, an interpretable PAN detail inject network (PDIN) is designed to inject the PAN details into the abundance feature efficiently. Comprehensive experiments on simulated and real datasets demonstrate the superiority and generality of our method over several state-of-the-art (SOTA) methods qualitatively and quantitatively (The codes in pytorch and paddle versions and dataset could be available at https://github.com/rs-lsl/Pgnet). (This is a improved version compared with the publication in Tgrs with the modification in the deduction of the PDIN block.)