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We study the spectrum of the normalized Laplace operator of a connected graph $Gamma$. As is well known, the smallest nontrivial eigenvalue measures how difficult it is to decompose $Gamma$ into two large pieces, whereas the largest eigenvalue controls how close $Gamma$ is to being bipartite. The smallest eigenvalue can be controlled by the Cheeger constant, and we establish a dual construction that controls the largest eigenvalue. Moreover, we find that the neighborhood graphs $Gamma[l]$ of order $lgeq2$ encode important spectral information about $Gamma$ itself which we systematically explore. In particular, the neighborhood graph method leads to new estimates for the smallest nontrivial eigenvalue that can improve the Cheeger inequality, as well as an explicit estimate for the largest eigenvalue from above and below. As applications of such spectral estimates, we provide a criterion for the synchronizability of coupled map lattices, and an estimate for the convergence rate of random walks on graphs.
The spectrum of the normalized graph Laplacian yields a very comprehensive set of invariants of a graph. In order to understand the information contained in those invariants better, we systematically investigate the behavior of this spectrum under lo
Determining and analyzing the spectra of graphs is an important and exciting research topic in theoretical computer science. The eigenvalues of the normalized Laplacian of a graph provide information on its structural properties and also on some rele
In this paper, using matrix techniques, we compute the Ihara-zeta function and the number of spanning trees of the join of two semi-regular bipartite graphs. Furthermore, we show that the spectrum and the zeta function of the join of two semi-regular bipartite graphs can determine each other.
A graph $G$ is $F$-saturated if it contains no copy of $F$ as a subgraph but the addition of any new edge to $G$ creates a copy of $F$. We prove that for $s geq 3$ and $t geq 2$, the minimum number of copies of $K_{1,t}$ in a $K_s$-saturated graph is
A graph $G$ contains $H$ as an emph{immersion} if there is an injective mapping $phi: V(H)rightarrow V(G)$ such that for each edge $uvin E(H)$, there is a path $P_{uv}$ in $G$ joining vertices $phi(u)$ and $phi(v)$, and all the paths $P_{uv}$, $uvin