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# A textbook of graph theory

17 Dec 1999-

TL;DR: In this article, direct-directed graphs are used to describe the properties of trees, independent sets and matchings, and Eulerian and Hamiltonian graphs for graph colouring.

Abstract: Basic Results.- Directed Graphs.- Connectivity.- Trees.- Independent Sets and Matchings.- Eulerian and Hamiltonian Graphs.- Graph Colourings.- Planarity.- Triangulated Graphs.- Applications.

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TL;DR: In this article, it was shown that for any positive integer n⩾3, there exist two equienergetic graphs of order 4n that are not cospectral.

919 citations

### Additional excerpts

...For notations and terminology, see [2,5]....

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TL;DR: This work focuses on anomaly detection in static graphs, which do not change and are capable of representing only a single snapshot of data, but as real‐world networks are constantly changing, there has been a shift in focus to dynamic graphs,Which evolve over time.

Abstract: Anomaly detection is an important problem with multiple applications, and thus has been studied for decades in various research domains. In the past decade there has been a growing interest in anomaly detection in data represented as networks, or graphs, largely because of their robust expressiveness and their natural ability to represent complex relationships. Originally, techniques focused on anomaly detection in static graphs, which do not change and are capable of representing only a single snapshot of data. As real-world networks are constantly changing, there has been a shift in focus to dynamic graphs, which evolve over time.

309 citations

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TL;DR: This is a bibliography of signed graphs and related mathematics, where work on weighted graphs are regarded as outside the scope of the bibliography — except (to some extent) when the author calls the weights "signs".

Abstract: A signed graph is a graph whose edges are labeled by signs. This is a bibliography of signed graphs and related mathematics. Several kinds of labelled graph have been called "signed" yet are mathematically very different. I distinguish four types: Group-signed graphs: the edge labels are elements of a 2-element group and are multiplied around a polygon (or along any walk). Among the natural generalizations are larger groups and vertex signs. Sign-colored graphs, in which the edges are labelled from a two-element set that is acted upon by the sign group: - interchanges labels, + leaves them unchanged. This is the kind of "signed graph" found in knot theory. The natural generalization is to more colors and more general groups — or no group. Weighted graphs, in which the edge labels are the elements +1 and -1 of the integers or another additive domain. Weights behave like numbers, not signs; thus I regard work on weighted graphs as outside the scope of the bibliography — ex cept (to some extent) when the author calls the weights "signs". Labelled graphs where the labels have no structure or properties but are called "signs" for any or no reason.

258 citations

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TL;DR: In this paper, a load restoration optimization model is proposed to coordinate topology reconfiguration and microgrid formation while satisfying a variety of operational constraints, which exploits benefits of operational flexibility provided by grid modernization to enable more critical load pickup.

232 citations

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TL;DR: A new model to reformulate the micorgrid formulation problem in resilient distribution networks is presented, such that the computational performance is significantly improved and the number of both binary and continuous variables is greatly reduced.

Abstract: Forming multiple micorgrids with distributed generators offers a resilient solution to restore critical loads from natural disasters in distribution systems. However, more dummy binary and continuous variables are needed with the increase of the number of microgrids, which will therefore increase the complexity of this model. To address this issue, this letter presents a new model to reformulate the micorgrid formulation problem in resilient distribution networks. Compared with the traditional model, the number of both binary and continuous variables is greatly reduced, such that the computational performance is significantly improved. Numerical results on IEEE test systems verify the effectiveness of the proposed model.

173 citations

### Cites background from "A textbook of graph theory"

...Furthermore, a sufficient and necessary condition for radiality is proposed in [5] that the graph is a radial if and only if the following two conditions are satisfied: (i) each sub-graph is a connected graph; (ii) the number of branches equals to the number of nodes minus the given number of sub-graphs....

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