Subgraphs and the Laplacian spectrum of a graph

AbstractLet G be a graph and H a subgraph of G. In this paper, a set of pairwise independent subgraphs that are all isomorphic copies of H is called an H-matching. Denoting by ν(H,G) the cardinality of a maximum H-matching in G, we investigate some relations between ν(H,G) and the Laplacian spectrum of G

On the Ring of Simultaneous Invariants for the Gleason–MacWilliams Group

AbstractWe construct a canonical generating set for the polynomial invariants of the simultaneous diagonal action (of arbitrary number of l factors) of the two-dimensional finite unitary reflection group G of order 192, which is called the group No. 9 in the list of Shephard and Todd, and is also called the Gleason–MacWilliams group. We find this canonical set in the vector space (⊗i=1lV)G, where V denotes the (dual of the) two-dimensional vector space on which the group G acts, by applying the techniques of Weyl (i.e., the polarization process of invariant theory) to the invariants C [ x, y ]G0of the two-dimensional group G0of order 48 which is the intersection of G and SL(2, C). It is shown that each element in this canonical set corresponds to an irreducible representation which appears in the decomposition of the action of the symmetric group Sl. That is, by letting the symmetric group Slacts on each element of the canonical generating set, we get an irreducible subspace on which the symmetric group Slacts irreducibly, and all these irreducible subspaces give the decomposition of the whole space (⊗i=1lV)G. This also makes it possible to find the generating set of the simultaneous diagonal action (of arbitrary l factors) of the group G. This canonical generating set is different from the homogeneous system of parameters of the simultaneous diagonal action of the group G. We can construct Jacobi forms (in the sense of Eichler and Zagier) in various ways from the invariants of the simultaneous diagonal action of the group G, and our canonical generating set is very fit and convenient for the purpose of the construction of Jacobi forms

Trojan Horse Method experiments with radioactive ion beams

The Trojan Horse Method (THM) is an indirect method that allows to get information about a two body reaction cross-section even at very low energy, avoiding the suppression effects due to the presence of the Coulomb barrier. The method requires a very accurate measurement of a three body reaction in order to reconstruct the whole kinematics and discriminate among different reaction mechanisms that can populate the same final state. These requirements hardly match with the typical low intensity and large divergence of radioactive ion beams (RIBs), and experimental improvements are mandatory for the applicability of the method. The first reaction induced by a radio activeion beam studied by applying the THM was the 18F(p,α)15O. Two experiments were performed in two different laboratories and using different experimental set-ups. The two experiments will be discussed and some results will be presented

Laplacian spectra and invariants of graphs

AbstractFor a connected graph G of order n⩾2 with positive Laplacian eigenvalues λ2,…,λn, letb(G)=n−11/λ2+⋯+1/λn.In this note we derive bounds on some graph invariants (edge-density in cuts, isoperimetric number, mean distance, edge-forwarding index, edge connectivity, etc) in terms of b(G)