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Structure and Representation
Published in Jonathan L. Gross, Jay Yellen, Mark Anderson, Graph Theory and Its Applications, 2018
Jonathan L. Gross, Jay Yellen, Mark Anderson
These two propositions lead to another simple proof of Euler’s theorem that the sum of the degrees equals twice the number of edges. In particular, the sum of the degrees is simply the sum of the row-sums of the incidence matrix, and the sum of the column-sums equals twice the number of edges. The result follows since these two sums both equal the sum of all the entries of the matrix.†
Expectation-Maximization Algorithm for Identification of Mesh-based Compartment Thermal Model of Power Modules
Published in Heat Transfer Engineering, 2023
Jakub Ševčík, Václav Šmídl, Ondřej Straka
The proposed compartment model given by equation (2) can be viewed as a particular case of directed graphs. Using graph theory [16], the heat transfer between compartments given by equation (2) may be described by a directed graph with vertices Ti and directed edges. Coefficients can be arbitrarily sorted into a vector , which corresponds to the ordering of edges in the graph. Then the directed graph can be represented by an incidence matrix. The incidence matrix is a sparse matrix of size n × m in general, where n is the number of vertices (i.e. compartments) and m is the number of edges (i.e. valid coefficients ). The element of the incidence matrix is defined by the relation
Dependability analysis of instrumented linear static systems based on their observability
Published in International Journal of Systems Science: Operations & Logistics, 2019
The matrix Cf of the fundamental cycles of the graph may be easily deduced from the incidence matrix. In the proposed example, using the expressions defining the nodes Ni (i=1, ..., 4), this incidence matrix is First, the incidence matrix A is partitioned (using permutation of columns) according to its regular part Ar :
The mise en scéne of memristive networks: effective memory, dynamics and learning
Published in International Journal of Parallel, Emergent and Distributed Systems, 2018
We now introduce a few mathematical definitions in order to clarify the discussion. Once an orientation has been assigned, and a set of oriented cycles is obtained, we can introduce two key matrices which will be used in the following: the directed incidence matrix , which is a matrix of size , and the cycle matrix , which is of size , where C is the number of cycles of the graph, M the number of edges and N the number of nodes. The incidence matrix has an entry if an (oriented) edge is leaving a node, if it is incoming to a node, and 0 otherwise. The directed incidence matrix labels edges on the rows and nodes on the columns: takes values is an edge is incoming on a node k, if it is outgoing, and 0 if the two are not incident. The cycle matrix has loop labels on its columns and edges on the rows: has entry if the directed edge is in the opposite direction of a chosen cycle , if it shares the same orientation, and 0 if it does not belong to that cycle. In what follows, we will assume that an orientation for the cycles and the currents have been chosen, as in Figure 2.