The fundamental relationship between matrices over the rational numbers and a newly defined type of graph, a Kirchhoff graph, is discussed. For a given matrix, a Kirchhoff graph for that matrix represents the orthogonal complementarity of the null and row spaces of the matrix. A number of basic results are proven, and then a relatively complicated Kirchhoff graph is constructed for a matrix tha...

The acoustic backscatter from pressure release prolate spheroids and a three-dimensional representation of a fish swimbladder (Chilean jack mackerel, Trachurus symmetricus murphyi) was calculated using four target strength models (Kirchhoff-approximation, Kirchhoff-ray-mode, finite element solution of the Helmholtz equation, and prolate-spheroid-modal-series). Smoothly varying errors were found...

where p and q are points on S, G is the free-space Green’s function, ∂/∂nq denotes normal differentiation at q, and uinc is the incident field. The Helmholtz integral equations are familiar boundary integral equations, for which there is a complete theoretical framework (Kleinman and Roach, 1974; Colton and Kress, 1983). Many examples of their numerical treatment, usually using boundary element...

The hyper-Kirchhoff index is introduced when the hyper-Wiener operator is applied to the resistance-distance matrix of a connected graph. We give lower and upper bounds for the hyper-Kirchhoff index, and determine the n-vertex unicyclic graphs with the smallest, the second and the third smallest as well as the largest, the second and the third largest hyper-Kirchhoff indices for n ≥ 5. We also ...

Abstract: We compare two mathematical theories that address duality between cycles and vertex-cuts of graphs in geometric settings. First, we propose a rigorous definition of a new type of graph, vector graphs. The special case of R2-vector graphs matches the intuitive notion of drawing graphs with edges taken as vectors. This leads to a discussion of Kirchhoff graphs, as originally presented b...

We consider the interaction of two vortex patches (elliptic Kirchhoff vortices) which move in an unbounded volume of an ideal incompressible fluid. A moment second-order model is used to describe the interaction. The case of integrability of a Kirchhoff vortex and a point vortex by the variable separation method is qualitatively analyzed. A new case of integrability of two Kirchhoff vortices is...

Inspired by a recent result of Davies and Pushnitski, we study resonance asymptotics of quantum graphs with general coupling conditions at the vertices. We derive a criterion for the asymptotics to be of a non-Weyl character. We show that for balanced vertices with permutation-invariant couplings the asymptotics is non-Weyl only in case of Kirchhoff or anti-Kirchhoff conditions. For graphs with...