An optimization approach to degree deviation and spectral radius

In this paper, we show the evaluation of the spectral radius for node degree as the basis to analyze the variation in the node degrees during the evolution of scale-free networks and small-world networks. Spectral radius is the principal eigenvalue of the adjacency matrix of a network graph and spectral radius ratio for node degree is the ratio of the spectral radius and the average node degree. We observe a very high positive correlation between the spectral radius ratio for node degree and the coefficient of variation of node degree (ratio of the standard deviation of node degree and average node degree). We show how the spectral radius ratio for node degree can be used as the basis to tune the operating parameters of the evolution models for scale-free networks and small-world networks as well as evaluate the impact of the number of links added per node introduced during the evolution of a scale-free network and evaluate the impact of the probability of rewiring during the evolution of a small-world network from a regular network.

A note on spectral radius and degree deviation in graphs

For a finite, simple, and undirected graph $G$ with $n$ vertices, $m$ edges, and largest eigenvalue $\lambda$, Nikiforov introduced the degree deviation of $G$ as$$s=\sum_{u\in V(G)}\left|d_G(u)-\frac{2m}{n}\right|.$$Contributing to a conjecture of Nikiforov, we show $\lambda-\frac{2m}{n}\leq \sqrt{\frac{2s}{3}}$. For our result, we show that the largest eigenvalue of a graph that arises from a bipartite graph with $m_{A,B}$ edges by adding $m_A$ edges within one of the two partite sets is at most $$\sqrt{m_A+m_{A,B}+\sqrt{m_A^2+2m_Am_{A,B}}},$$which is a common generalization of results due to Stanley and Bhattacharya, Friedland, and Peled.