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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.
The subdivision graph $mathcal{S}(G)$ of a graph $G$ is the graph obtained by inserting a new vertex into every edge of $G$. Let $G_1$ and $G_2$ be two vertex disjoint graphs. The emph{subdivision-vertex join} of $G_1$ and $G_2$, denoted by $G_1dot{v
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 contro
Let $mathrm{rex}(n, F)$ denote the maximum number of edges in an $n$-vertex graph that is regular and does not contain $F$ as a subgraph. We give lower bounds on $mathrm{rex}(n, F)$, that are best possible up to a constant factor, when $F$ is one of
Two method for computation of the spectra of certain infinite graphs are suggested. The first one can be viewed as a reversed Gram--Schmidt orthogonalization procedure. It relies heavily on the spectral theory of Jacobi matrices. The second method is
For hyperbolic Riemann surfaces of finite geometry, we study Selbergs zeta function and its relation to the relative scattering phase and the resonances of the Laplacian. As an application we show that the conjugacy class of a finitely generated, tor