No abstract

The graph connectivity is a fundamental concept in graph theory. In particular, it plays a vital role in applications related to the modern interconnection graphs, e.g., it can be used to measure the vulnerability of the corresponding graph, and is an important metric for reliability and fault tolerance of the graph. Here, firstly, we introduce two types of divided operations, named vertex-divided operation and edge-divided operation, respectively, as well as their inverse operations vertex-coincident operation and edge-coincident operation, to find some methods for splitting vertices of graphs. Secondly, we define a new connectivity, which can be referred to as divided connectivity, which differs from traditional connectivity, and present an equivalence relationship between traditional connectivity and our divided connectivity. Afterwards, we explore the structures of graphs based on the vertex-divided connectivity. Then, as an application of our divided operations, we show some necessary and sufficient conditions for a graph to be an Euler’s graph. Finally, we propose some valuable and meaningful problems for further research.

For random walks on a complex network, the configuration of a network that provides optimal or suboptimal navigation efficiency is meaningful research. It has been proven that a complete graph has the exact minimal mean hitting time, which grows linearly with the network order. In this paper, we present a class of sparse networks G(t) in view of a graphic operation, which have a similar dynamic process with the complete graph; however, their topological properties are different. We capture that G(t) has a remarkable scale-free nature that exists in most real networks and give the recursive relations of several related matrices for the studied network. According to the connections between random walks and electrical networks, three types of graph invariants are calculated, including regular Kirchhoff index, M-Kirchhoff index and A-Kirchhoff index. We derive the closed-form solutions for the mean hitting time of G(t), and our results show that the dominant scaling of which exhibits the same behavior as that of a complete graph. The result could be considered when designing networks with high navigation efficiency.