On the extremal properties of the average eccentricity

The eccentricity of a vertex is the maximum distance from it to another vertex and the average eccentricity $ecc (G)$ of a graph $G$ is the mean value of eccentricities of all vertices of $G$. The average eccentricity is deeply connected with a topological descriptor called the eccentric connectivity index, defined as a sum of products of vertex degrees and eccentricities. In this paper we analyze extremal properties of the average eccentricity, introducing two graph transformations that increase or decrease $ecc (G)$. Furthermore, we resolve four conjectures, obtained by the system AutoGraphiX, about the average eccentricity and other graph parameters (the clique number, the Randi\' c index and the independence number), refute one AutoGraphiX conjecture about the average eccentricity and the minimum vertex degree and correct one AutoGraphiX conjecture about the domination number.Comment: 15 pages, 3 figure

On Domination Number and Distance in Graphs

A vertex set $S$ of a graph $G$ is a \emph{dominating set} if each vertex of $G$ either belongs to $S$ or is adjacent to a vertex in $S$. The \emph{domination number} $\gamma(G)$ of $G$ is the minimum cardinality of $S$ as $S$ varies over all dominating sets of $G$. It is known that $\gamma(G) \ge \frac{1}{3}(diam(G)+1)$, where $diam(G)$ denotes the diameter of $G$. Define $C_r$ as the largest constant such that $\gamma(G) \ge C_r \sum_{1 \le i < j \le r}d(x_i, x_j)$ for any $r$ vertices of an arbitrary connected graph $G$; then $C_2=\frac{1}{3}$ in this view. The main result of this paper is that $C_r=\frac{1}{r(r-1)}$ for $r\geq 3$. It immediately follows that $\gamma(G)\geq \mu(G)=\frac{1}{n(n-1)}W(G)$, where $\mu(G)$ and $W(G)$ are respectively the average distance and the Wiener index of $G$ of order $n$. As an application of our main result, we prove a conjecture of DeLaVi\~{n}a et al.\;that $\gamma(G)\geq \frac{1}{2}(ecc_G(B)+1)$, where $ecc_G(B)$ denotes the eccentricity of the boundary of an arbitrary connected graph $G$.Comment: 5 pages, 2 figure

Models for On-line Social Networks

On-line social networks such as Facebook or Myspace are of increasing interest to computer scientists, mathematicians, and social scientists alike. In such real-world networks, nodes represent people and edges represent friendships between them. Mathematical models have been proposed for a variety of complex real-world networks such as the web graph, but relatively few models exist for on-line social networks. We present two new models for on-line social networks: a deterministic model we call Iterated Local Transitivity (ILT), and a random ILT model. We study various properties in the deterministic ILT model such as average degree, average distance, and diameter. We show that the domination number and cop number stay the same no matter how many nodes or edges are added over time. We investigate the automorphism groups and eigenvalues of graphs generated by the ILT model. We show that the random ILT model follows a power-law degree distribution and we provide a theorem about the power law exponent of this model. We present simulations for the degree distribution of the random ILT model

Multiple domination models for placement of electric vehicle charging stations in road networks

Electric and hybrid vehicles play an increasing role in the road transport networks. Despite their advantages, they have a relatively limited cruising range in comparison to traditional diesel/petrol vehicles, and require significant battery charging time. We propose to model the facility location problem of the placement of charging stations in road networks as a multiple domination problem on reachability graphs. This model takes into consideration natural assumptions such as a threshold for remaining battery load, and provides some minimal choice for a travel direction to recharge the battery. Experimental evaluation and simulations for the proposed facility location model are presented in the case of real road networks corresponding to the cities of Boston and Dublin.Comment: 20 pages, 5 figures; Original version from March-April 201