On the Extra Connectivity of the Power Graph of a Finite Cyclic Group

On Spectral radius and second largest eigenvalue of power graphs of finite groups

Consider a group [Formula: see text] and construct its power graph, whose vertex set consists of the elements of [Formula: see text]. Two distinct vertices (elements) are adjacent in the graph if and only if one element can be expressed as an integral power of the other. In this article, we improved the bounds of the spectral radius of the power graphs of the cyclic group [Formula: see text], the dihedral group [Formula: see text], and the dicyclic group [Formula: see text]. For [Formula: see text] the power graph of the cyclic group [Formula: see text] is not a complete multipartite graph. We find the second largest eigenvalue bounds of the same with the clique number. In some cases, we find the bounds are exact if and only if they belong to a particular family of graphs. Lastly, we work on the distance spectral radius of the power graphs of the same groups

A Note on Short-Range Network Communication and Clustering

Complex systems of interacting components often can be modeled by a graph that consists of a set of n nodes and a set of m edges. Such a graph can be represented by an adjacency matrix A∈Rn×n, whose (ij)th entry is one if there is an edge pointing from node i to node j, and is zero otherwise. The matrix A and its low-order powers reveal important properties of the graph and allow the enumeration of short paths and cycles that are important for determining short-range communication in the graph as well as node clustering. Closed-form expressions for path matrices of length up to four are derived, and a novel indicator of the structural propensity of the graph to form clusters is proposed. Numerical examples illustrate our analysis.

Protecting the Connectivity of a Graph Under Nonuniform Edge Failures

Abstract. We study the problem of guaranteeing the connectivity of a given graph by protecting or strengthening edges. Herein, a protected edge is assumed to be robust and will not fail, which features a nonuniform failure model. We introduce the [Formula: see text]-Steiner-Connectivity Preservation problem where we protect a minimum-cost set of edges such that the underlying graph maintains [Formula: see text]-edge-connectivity between given terminal pairs against edge failures, assuming at most [Formula: see text] unprotected edges can fail. We design polynomial-time exact algorithms for the cases where [Formula: see text] and [Formula: see text] are small and approximation algorithms for general values of [Formula: see text] and [Formula: see text]. Additionally, we show that when both [Formula: see text] and [Formula: see text] are part of the input, even deciding whether a given solution is feasible is [Formula: see text]-complete. This hardness also carries over to Flexible Network Design, a research direction that has gained significant attention. In particular, previous work focuses on problem settings where either [Formula: see text] or [Formula: see text] is constant, for which our new hardness result now provides justification.

Theory & Applications of Graphs Theory & Applications of Graphs

Hierarchical One-Link Interconnection Networks for Low-Degree Parallel Communication

The results support HON as a simple low-degree construction for structured inter-group communication, whereas higher-radix, adaptive, or more richly connected fabrics remain better suited to less structured traffic and larger bandwidth demand.

Adjacency spectrum of power graphs of Z m × Z n