Generalized exponents of non-primitive graphs

AbstractThe exponent of a primitive digraph is the smallest integer k such that for each ordered pair of (not necessarily distinct) vertices x and y there is a walk of length k from x to y. As a generalization of exponent, Brualdi and Liu (Linear Algebra Appl. 14 (1990) 483–499) introduced three types of generalized exponents for primitive digraphs in 1990. In this paper we extend their definitions of generalized exponents from primitive digraphs to general digraphs which are not necessarily primitive. We give necessary and sufficient conditions for the finiteness of these generalized exponents for graphs (undirected, corresponding to symmetric digraphs) and completely determine the largest finite values and the exponent sets of generalized exponents for the class of non-primitive graphs of order n, the class of connected bipartite graphs of order n and the class of trees of order n

Some results on the ordering of the Laplacian spectral radii of unicyclic graphs

AbstractA unicyclic graph is a graph whose number of edges is equal to the number of vertices. Guo Shu-Guang [S.G. Guo, The largest Laplacian spectral radius of unicyclic graph, Appl. Math. J. Chinese Univ. Ser. A. 16 (2) (2001) 131–135] determined the first four largest Laplacian spectral radii together with the corresponding graphs among all unicyclic graphs on n vertices. In this paper, we extend this ordering by determining the fifth to the ninth largest Laplacian spectral radii together with the corresponding graphs among all unicyclic graphs on n vertices

Ray solvable linear systems and ray S2NS matrices

AbstractRay solvable linear systems and ray S2NS matrices are complex generalizations of the sign solvable linear systems and S2NS matrices. We use the determinantal ray unique matrices (instead of ray nonsingular matrices) as a generalization of SNS matrices, to generalize some fundamental results of S2NS matrices from the real case to complex case, such as the graph theoretical characterization, the inverse ray patterns and the upper bound of the number of nonzero entries of S2NS matrices. The well known characterization of the sign solvable linear systems (in terms of the L-matrices and S∗ matrices) is also generalized to ray solvable linear systems, and the relationships between the ray S∗-matrices and real S∗-matrices are investigated. Some examples are also given to illustrate that some results, such as the characterization of the sign inconsistent linear systems, do not carry over to the complex case

Vehicle Path Planning with Multicloud Computation Services

With the development of artificial intelligence, public cloud service platforms have begun to provide common pretrained object recognition models for public use. In this study, a dynamic vehicle path-planning system is developed, which uses several general pretrained cloud models to detect obstacles and calculate the navigation area. The Euclidean distance and the inequality based on the detected marker box data are used for vehicle path planning. Experimental results show that the proposed method can effectively identify the driving area and plan a safe route. The proposed method integrates the bounding box information provided by multiple cloud object detection services to detect navigable areas and plan routes. The time required for cloud-based obstacle identification is 2 s per frame, and the time required for feasible area detection and action planning is 0.001 s per frame. In the experiments, the robot that uses the proposed navigation method can plan routes successfully

An edge grafting theorem on the energy of unicyclic and bipartite graphs

AbstractThe energy of a graph is the sum of the absolute values of the eigenvalues of its adjacency matrix. The edge grafting operation on a graph is certain kind of edge moving between two pendant paths starting from the same vertex. In this paper we show how the graph energy changes under the edge grafting operations on unicyclic and bipartite graphs. We also give some applications of this result on the comparison of graph energies between unicyclic or bipartite graphs