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Journal ArticleDOI

Distance Estrada index of random graphs

04 Mar 2015-Linear & Multilinear Algebra (Taylor & Francis)-Vol. 63, Iss: 3, pp 466-471
TL;DR: In this paper, lower and upper bounds for the distance Estrada index of a simple graph were established for almost all graphs, and the eigenvalues of its distance matrix were derived.
Abstract: Suppose is a simple graph and are the eigenvalues of its distance matrix . The distance Estrada index of is defined as the sum of , . In this paper, we establish better lower and upper bounds to for almost all graphs .
Citations
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Proceedings ArticleDOI
22 Jan 2006
TL;DR: Some of the major results in random graphs and some of the more challenging open problems are reviewed, including those related to the WWW.
Abstract: We will review some of the major results in random graphs and some of the more challenging open problems. We will cover algorithmic and structural questions. We will touch on newer models, including those related to the WWW.

7,116 citations

Journal ArticleDOI
Yilun Shang1
30 Mar 2015-PLOS ONE
TL;DR: It is found that neither the static snapshot graphs nor the aggregated graph can approximate the evolving graph itself, indicating the fundamental difference between the static and dynamic Estrada indices.
Abstract: Large-scale time-evolving networks have been generated by many natural and technological applications, posing challenges for computation and modeling. Thus, it is of theoretical and practical significance to probe mathematical tools tailored for evolving networks. In this paper, on top of the dynamic Estrada index, we study the dynamic Laplacian Estrada index and the dynamic normalized Laplacian Estrada index of evolving graphs. Using linear algebra techniques, we established general upper and lower bounds for these graph-spectrum-based invariants through a couple of intuitive graph-theoretic measures, including the number of vertices or edges. Synthetic random evolving small-world networks are employed to show the relevance of the proposed dynamic Estrada indices. It is found that neither the static snapshot graphs nor the aggregated graph can approximate the evolving graph itself, indicating the fundamental difference between the static and dynamic Estrada indices.

28 citations

Journal ArticleDOI
TL;DR: In this article, the generalized distance matrix of a simple connected graph G is defined as a convex combination of the convex combinations of T r (G ) + (1 − α ) D (G) for 0 ≤ α ≤ 1.

21 citations

Journal ArticleDOI
13 Jun 2019-Energies
TL;DR: In this article, the authors employed hierarchical cluster analysis in an attempt to distinguish those countries among the new EU Member States that increased their electrical capacity from renewable energy sources to the greatest extent while paying attention to their energy intensity.
Abstract: Climate change and awareness of the need to care for the environment have resulted in a global increase in the interest in renewable energy sources. The European Union (EU) is active in this respect and requires Member States to fulfill specific plans in the transformation of their energy systems. We employed hierarchical cluster analysis in an attempt to distinguish those countries among the new EU Member States that increased their electrical capacity from renewable energy sources to the greatest extent while paying attention to their energy intensity. The analyses were conducted in two scenarios for both 2004 and 2016. The first scenario assumed an analysis of all known renewable energy sources, whereas in the second scenario, only renewable energy sources from wind and solar power plants were included. The division of analyses into these two variants showed the importance of the differences in the energy assessment of individual countries, depending on classification of renewable energy sources. We identified groups of countries where electrical capacity from renewable energy sources increased the most. Conducting analyses using two variants allowed distinguishing countries that based most of their renewable energy on modern renewable energy sources, such as solar and wind power plants. The inclusion of gross domestic product in the analyses allowed us to identify countries with the worst energy efficiency value.

12 citations

Journal ArticleDOI
19 Oct 2019
TL;DR: In this paper, the generalized distance Estrada index of a graph G is defined as a generalized distance matrix D α (G ) = ∑ i = 1 n e ∂ i − 2 α W ( G ) n, where W denotes the Wiener index of G.
Abstract: Suppose that G is a simple undirected connected graph. Denote by D ( G ) the distance matrix of G and by T r ( G ) the diagonal matrix of the vertex transmissions in G, and let α ∈ [ 0 , 1 ] . The generalized distance matrix D α ( G ) is defined as D α ( G ) = α T r ( G ) + ( 1 − α ) D ( G ) , where 0 ≤ α ≤ 1 . If ∂ 1 ≥ ∂ 2 ≥ … ≥ ∂ n are the eigenvalues of D α ( G ) ; we define the generalized distance Estrada index of the graph G as D α E ( G ) = ∑ i = 1 n e ∂ i − 2 α W ( G ) n , where W ( G ) denotes for the Wiener index of G. It is clear from the definition that D 0 E ( G ) = D E E ( G ) and 2 D 1 2 E ( G ) = D Q E E ( G ) , where D E E ( G ) denotes the distance Estrada index of G and D Q E E ( G ) denotes the distance signless Laplacian Estrada index of G. This shows that the concept of generalized distance Estrada index of a graph G merges the theories of distance Estrada index and the distance signless Laplacian Estrada index. In this paper, we obtain some lower and upper bounds for the generalized distance Estrada index, in terms of various graph parameters associated with the structure of the graph G, and characterize the extremal graphs attaining these bounds. We also highlight relationship between the generalized distance Estrada index and the other graph-spectrum-based invariants, including generalized distance energy. Moreover, we have worked out some expressions for D α E ( G ) of some special classes of graphs.

12 citations

References
More filters
Proceedings ArticleDOI
22 Jan 2006
TL;DR: Some of the major results in random graphs and some of the more challenging open problems are reviewed, including those related to the WWW.
Abstract: We will review some of the major results in random graphs and some of the more challenging open problems. We will cover algorithmic and structural questions. We will touch on newer models, including those related to the WWW.

7,116 citations


"Distance Estrada index of random gr..." refers methods in this paper

  • ...Lemma 3 [19] Suppose that p2n − 2 ln n → ∞ and n2(1 − p) → ∞ as n → ∞....

    [...]

  • ...Preliminaries To begin with, we recall the Erdős–Rényi random graph Gn(p),[19] which consists of all graphs on n vertices in which the edges are chosen independently with probability p = p(n) ∈ (0, 1)....

    [...]

  • ...s.[19] We have √ n2 + 4m e2(n−1)−np−O( √ n) = √ (2p + 1)n2 − 2pn e2(n−1)−np−O( √ n) = o(1) a....

    [...]

Book
01 Jan 1995
TL;DR: The Spectrum and the Group of Automorphisms as discussed by the authors have been used extensively in Graph Spectra Techniques in Graph Theory and Combinatory Applications in Chemistry an Physics. But they have not yet been applied to Graph Spectral Biblgraphy.
Abstract: Introduction. Basic Concepts of the Spectrum of a Graph. Operations on Graphs and the Resulting Spectra. Relations Between Spectral and Structural Properties of Graphs. The Divisor of a Graph. The Spectrum and the Group of Automorphisms. Characterization of Graphs by Means of Spectra. Spectra Techniques in Graph Theory and Combinatories. Applications in Chemistry an Physics. Some Additional Results. Appendix. Tables of Graph Spectra Biblgraphy. Index of Symbols. Index of Names. Subject Index.

2,119 citations

MonographDOI
01 Jan 2001

2,064 citations

Book
01 Jan 1990

1,185 citations