The Minimal Spectral Radius of Graphs with a Given Diameter

AMS classsifications: 05C50; 05E99; 94C15;graphs;spectral radius;diameter;networks;virus propagation

The minimal spectral radius of graphs with a given diameter

AbstractThe spectral radius of a graph (i.e., the largest eigenvalue of its corresponding adjacency matrix) plays an important role in modeling virus propagation in networks. In fact, the smaller the spectral radius, the larger the robustness of a network against the spread of viruses. Among all connected graphs on n nodes the path Pn has minimal spectral radius. However, its diameter D, i.e., the maximum number of hops between any pair of nodes in the graph, is the largest possible, namely D=n−1. In general, communication networks are designed such that the diameter is small, because the larger the number of nodes traversed on a connection, the lower the quality of the service running over the network. This leads us to state the following problem: which connected graph on n nodes and a given diameter D has minimal spectral radius? In this paper we solve this problem explicitly for graphs with diameter D∈1,2,n2,n-3,n-2,n-1. Moreover, we solve the problem for almost all graphs on at most 20 nodes by a computer search

Real Polynomial Systems of Degree n with n + 1 Line Invariants

AbstractIn this paper we study the number of limit cycles for special classes of real polynomial systems of differential equations on the plane. It is shown that for polynomial systems of degree n which possess n + 1 line invariants both the relative positions of the n + 1 line invariants and their types (i.e., real or complex) influence the number of limit cycles

Effective graph resistance

AbstractThis paper studies an interesting graph measure that we call the effective graph resistance. The notion of effective graph resistance is derived from the field of electric circuit analysis where it is defined as the accumulated effective resistance between all pairs of vertices. The objective of the paper is twofold. First, we survey known formulae of the effective graph resistance and derive other representations as well. The derivation of new expressions is based on the analysis of the associated random walk on the graph and applies tools from Markov chain theory. This approach results in a new method to approximate the effective graph resistance. A second objective of this paper concerns the optimisation of the effective graph resistance for graphs with given number of vertices and diameter, and for optimal edge addition. A set of analytical results is described, as well as results obtained by exhaustive search. One of the foremost applications of the effective graph resistance we have in mind, is the analysis of robustness-related problems. However, with our discussion of this informative graph measure we hope to open up a wealth of possibilities of applying the effective graph resistance to all kinds of networks problems

Modeling ping times in first person shooter games

In First Person Shooter (FPS) games the Round Trip Time (RTT), i.e., the sum of the network delay from client to server and the network delay from server to client, impacts the game

Performance of TCP with multiple Priority Classes

We consider the dimensioning problem for Internet access links carrying TCP traffic with two priority classes. To this end, we study the behaviour of TCP at the flow level described by a multiple-server Processor Sharing (PS) queueing model with two customer classes, where the customers represent flows generated by downloading Internet objects; the sojourn times represent the object transfer times. We present closed-form expressions for the mean sojourn times for high-priority customers and approximate expressions for the mean sojourn times of low-priority customers. The accuracy of the model is demonstrated by comparing results based on the PS model with "real" TCP simulation results obtained by the well-known Network Simulator. The experimental results demonstrate that the model-based results are highly accurate when the mean object size is at least 10 IP-packets, and the loss rate is negligible

A quantum algorithm for minimising the effective graph resistance upon edge addition

In this work, we consider the following problem: given a graph, the addition of which single edge minimises the effective graph resistance of the resulting (or, augmented) graph. A graph’s effective graph resistance is inversely proportional to its robustness, which means the graph augmentation problem is relevant to, in particular, applications involving the robustness and augmentation of complex networks. On a classical computer, the best known algorithm for a graph with N vertices has time complexity (Formula Presented). We show that it is possible to do better: Dürr and Høyer’s quantum algorithm solves the problem in time (Formula Presented). We conclude with a simulation of the algorithm and solve ten small instances of the graph augmentation problem on the Quantum Inspire quantum computing platform