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Abstract: The Szeged index extends the Wiener index for cyclic graphs by counting the number of vertices on both sides of each edge and sum these counts. Klavzar et al. [S. Klavzar, A. Rajapakse, I. Gutman, The Szeged and the Wiener index of graphs, Appl. Math. Lett. 9 (5) (1996) 45–49] provided an exact formula for computing Szeged index of product of graphs. In this paper, we apply a matrix method to obtain exact formulae for computing the Szeged index of join and composition of graphs. The join and composition of the vertex PI index of graphs are also computed. [Copyright &y& Elsevier]

Abstract: A tree which has exactly one vertex of degree greater than two is said to be starlike. In spite of seemingly simple structure of these trees, not much is known about their spectral properties. In this paper, we introduce a generalization of the notion of cospectrality called -cospectrality which turns out to be useful in constructing cospectral graphs. Based on this, we construct cospectral mates for some starlike trees. We also present a set of necessary and sufficient conditions for divisibility of the characteristic polynomial of a starlike tree by the characteristic polynomial of a path. [Copyright &y& Elsevier]

We model the folding of ordinary paper via piecewise isometries R2→R3. The collection of crease lines and vertices in the unfolded paper is called the crease pattern. Our results generalize the previously known necessity conditions from the more restrictive case of folding paper flat (into R2); if the crease pattern is foldable, then the product (in a non-intuitive order) of the associated rotational matrices is the identity matrix. This condition holds locally in a multiple vertex crease pattern and can be adapted to a global condition. Sufficiency conditions are significantly harder, and are not known except in the two-dimensional single-vertex case. [Copyright &y& Elsevier]