The energy of a graph is the sum of the singular values of its adjacency matrix. We are interested in how the energy of a graph changes when edges are deleted. Examples show that all cases are possible: increased, decreased, unchanged. Our goal is to find possible graph theoretical descriptions and to provide an infinite family of graphs for each case. The main tool is a singular value inequali...

Model reduction refers to reducing the complexity of a given model to achieve a balance between model simplicity and accuracy. This chapter presents a set of model reduction techniques that are particularly amenable to bond graph models due to the common energy-based nature of these techniques and the bond graph language. Three techniques are presented that are developed with model order reduct...

In this paper, the concept of fuzzy graph is extended to a complex fuzzy graph. Complex fuzzy graphs are encountered in complex fuzzy set theory. The novelty of the complex fuzzy set lies in the range of values its membership function may attain. In contrast to a traditional fuzzy membership function, this range is not limited to [0, 1], but extended to the unit circle in the complex plane. Thu...

Nonlinear functionals that appear as a product of two integrals are here considered in the context of elastic curves of variable length. A technique is introduced that exploits the fact that one of the integrals has an integrand independent of the derivative of the unknown. Both the linear and the nonlinear cases are illustrated. By lengthening parameterized curves it is possible to reduce the ...

The energy of a graph G is equal to the sum of the absolute values of the eigenvalues of G. We define the matching energy (ME) of the graph G as the sum of the absolute values of the zeros of the matching polynomial of G, and determine its basic properties. It is pointed out that the chemical applications of ME go back to the 1970s.

We introduce an energy model whose minimum energy drawings reveal the clusters of the drawn graph. Here a cluster is a set of nodes with many internal edges and few edges to nodes outside the set. The drawings of the bestknown force and energy models do not clearly show clusters for graphs whose diameter is small relative to the number of nodes. We formally characterize the minimum energy drawi...

We discuss several criteria for clustering graphs, and identify two criteria which are not biased towards certain cluster sizes: the node-normalized cut (also called cut ratio) and the edge-normalized cut. We present two energy models whose minimum energy drawings reveal clusters with respect to these criteria. The energy model that corresponds to the edge-normalized cut differs from the other ...

Let G be a vertex colored graph. The minimum number χ(G) of colors needed for coloring of a graph G is called the chromatic number. Recently, Adiga et al. [1] have introduced the concept of color energy of a graph Ec(G) and computed the color energy of few families of graphs with χ(G) colors. In this paper we derive explicit formulas for the color energies of the unitary Cayley graph Xn, the co...

In this paper we address the problem of minimizing a large class of energy functions that occur in early vision. The major restriction is that the energy function’s smoothness term must only involve pairs of pixels. We propose two algorithms that use graph cuts to compute a local minimum even when very large moves are allowed. The first move we consider is an α-βswap: for a pair of labels α, β,...