Combinatorial Perron parameters for trees

A Combinatorial Theorem for Trees

Unhooking Circulant Graphs: A Combinatorial Method for Counting Spanning Trees and Other Parameters

It has long been known that the number of spanning trees in circulant graphs with fixed jumps and n nodes satisfies a recurrence relation in n. The proof of this fact was algebraic (relating the products of eigenvalues of the graphs’ adjacency matrices) and not combinatorial. In this paper we derive a straightforward combinatorial proof of this fact. Instead of trying to decompose a large circu...

A Combinatorial Approach for Analyzing the Number of Descendants in Increasing Trees and Related Parameters

This work is devoted to a study of the number of descendants of node j in random increasing trees, previously treated in [4, 6, 8, 13], and also to a study of a more general quantity called generalized descendants, which arises in a certain growth process. Our study is based on a combinatorial approach by establishing a bijection with certain lattice paths. For the parameters considered we deri...

Quantitative Combinatorial Geometry for Continuous Parameters

Tensor algebras and combinatorial parameters

For any real inner product space V , let T (V ) denote its tensor algebra. We give a theorem giving conditions under which a real-valued algebra homomorphism from certain subalgebras of T (V ) can be extended to T (V ). The main applications are characterizations of combinatorial parameters by means of ‘reflection positivity’. The proof method follows mainly that of Szegedy [4], whose theorem i...