In this paper we study various models for web graphs with respect to bounded expansion. All the deterministic models even have constant expansion, whereas the copying model has unbounded expansion. The most interesting case turns out to be the preferential attachment model --- which we conjecture to have unbounded expansion, too.

For the notion of neighbor-connectivity in graphs whenever a vertex is subverted the entire closed neighborhood of the vertex is deleted from the graph. The minimum number of vertices whose subversion results in an empty, complete, or disconnected subgraph is called the neighbor-connectivity of the graph. Gunther, Hartnell, and Nowakowski have shown that for any graph, neighbor-connectivity is bounded above by κ. Doty has sharpened that bound in abelian Cayley graphs to approximately (1/2)κ. The...

P. Kristiansen, S. M. Hedetniemi, and S. T. Hedetniemi, in Alliances in graphs, J. Combin. Math. Combin. Comput. 48 (2004), 157–177, and T. W. Haynes, S. T. Hedetniemi, and M. A. Henning, in Global defensive alliances in graphs, Electron. J. Combin. 10 (2003), introduced the defensive alliance number $a\left(G\right)$, strong defensive alliance number $\widehat{a}\left(G\right)$, and global defensive alliance number ${\gamma }_a\left(G\right)$. In this paper, we consider relationships between these parameters and the domination number $\gamma \left(G\right)$. For any positive integers...

If a graph is connected then the largest eigenvalue (i.e., index) generally changes (decreases or increases) if some local modifications are performed. In this paper two types of modifications are considered: (i) for a fixed vertex, t edges incident with it are deleted, while s new edges incident with it are inserted; (ii) for two non-adjacent vertices, t edges incident with one vertex are deleted, while s new edges incident with the other vertex are inserted. ...

A b-coloring of a graph G with k colors is a proper coloring of G using k colors in which each color class contains a color dominating vertex, that is, a vertex which has a neighbor in each of the other color classes. The largest positive integer k for which G has a b-coloring using k colors is the b-chromatic number b(G) of G. In this paper, we obtain bounds for the b- chromatic number of induced subgraphs in terms of the b-chromatic number of the original graph. This turns out to be a generalization...

Let $G$ be a simple connected graph of order $n$ with degree sequence $(d_1,d_2,...,d_n)$. Denote ${{(}^{\alpha }t)}_i={\sum }_{j:i\sim j}{d}_j^{\alpha }$, ${{(}^{\alpha }m)}_i={{(}^{\alpha }t)}_i/d_i^{\alpha }$ and ${{(}^{\alpha }N)}_i={\sum }_{j:i\sim j}{{(}^{\alpha }t)}_j$, where $\alpha$ is a real number. Denote by ${\lambda }_1\left(G\right)$ and ${\mu }_1\left(G\right)$ the spectral radius of the adjacency matrix and the Laplacian matrix of $G$, respectively. In this paper, we present some upper and lower bounds of ${\lambda }_1\left(G\right)$ and ${\mu }_1\left(G\right)$ in terms of ${{(}^{\alpha }t)}_i$, ${{(}^{\alpha }m)}_i$ and ${{(}^{\alpha }N)}_i$. Furthermore, we also characterize some extreme graphs which attain these upper bounds. These results theoretically improve and generalize some known results.