2,072 research outputs found
Graph measures and network robustness
Network robustness research aims at finding a measure to quantify network
robustness. Once such a measure has been established, we will be able to
compare networks, to improve existing networks and to design new networks that
are able to continue to perform well when it is subject to failures or attacks.
In this paper we survey a large amount of robustness measures on simple,
undirected and unweighted graphs, in order to offer a tool for network
administrators to evaluate and improve the robustness of their network. The
measures discussed in this paper are based on the concepts of connectivity
(including reliability polynomials), distance, betweenness and clustering. Some
other measures are notions from spectral graph theory, more precisely, they are
functions of the Laplacian eigenvalues. In addition to surveying these graph
measures, the paper also contains a discussion of their functionality as a
measure for topological network robustness
Sharp Bounds on (Generalized) Distance Energy of Graphs
Given a simple connected graph G, let D(G) be the distance matrix, DL(G) be the distance Laplacian matrix, DQ(G) be the distance signless Laplacian matrix, and Tr(G) be the vertex transmission diagonal matrix of G. We introduce the generalized distance matrix Dα(G)=αTr(G)+(1−α)D(G) , where α∈[0,1] . Noting that D0(G)=D(G),2D12(G)=DQ(G),D1(G)=Tr(G) and Dα(G)−Dβ(G)=(α−β)DL(G) , we reveal that a generalized distance matrix ideally bridges the spectral theories of the three constituent matrices. In this paper, we obtain some sharp upper and lower bounds for the generalized distance energy of a graph G involving different graph invariants. As an application of our results, we will be able to improve some of the recently given bounds in the literature for distance energy and distance signless Laplacian energy of graphs. The extremal graphs of the corresponding bounds are also characterized
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