Sharp Upper Bounds for Energy and Randic Energy
01 Jan 2013-
TL;DR: In this paper, sharp upper bounds for the energy and Randic energy of a bipartite graph were established, from which some previously known results could be deduced, and some previously unknown results could also be inferred.
Abstract: Sharp upper bounds for the energy and Randic energy of a (bipartite) graph are established. From these, some previously known results could be deduced.
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TL;DR: This work pursues with this line of research by proving further extremal properties of the degree-based graph entropies by proving their extremal values when considered Shannon entropy- based graph measures.
57 citations
TL;DR: In this article, the energy of the general extended adjacency matrix A = (a i j ) of a simple graph of order n and size m is defined as a i j = F (d i, d j ) if the vertices v i and v j are adjacent, and a I j = 0 otherwise.
41 citations
Posted Content•
TL;DR: This article describes the graphs with minimal ME among all unicyclic and bicyclic graphs with a given diameter d and defines the roots of its matching polynomial.
Abstract: Gutman and Wagner proposed the concept of matching energy (ME) and pointed out that the chemical applications of ME go back to the 1970s. Let $G$ be a simple graph of order $n$ and $\mu_1,\mu_2,\ldots,\mu_n$ be the roots of its matching polynomial. The matching energy of $G$ is defined to be the sum of the absolute values of $\mu_{i}\ (i=1,2,\ldots,n)$. In this paper, we characterize the graphs with minimal matching energy among all unicyclic and bicyclic graphs with a given diameter $d$.
40 citations
Cites background from "Sharp Upper Bounds for Energy and R..."
...There are various generalizations of graph energy, such as Randić energy [3, 10], Laplacian energy [7], distance energy [36], incidence energy [4, 5], energy of matrices [14] and energy of a polynomial [33], etc....
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TL;DR: The necessary and sufficient conditions for the positive of a mixed graph are presented and the incident matrix and Hermitian Laplacian matrix of a Mixed graph are introduced and some results about the Hermitia LaplACian spectrum are derived.
27 citations
TL;DR: The explicit asymptotic values of Laplacian energies for various lattices are obtained and the various topological indices per vertex of lattice systems are independent of the toroidal, cylindrical, and free boundary conditions.
23 citations
References
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Book•
01 Aug 1979
TL;DR: 1. Matrices which leave a cone invariant 2. Nonnegative matrices 3. Semigroups of non negative matrices 4. Symmetric nonnegativeMatrices 5. Generalized inverse- Positivity 6. M-matrices 7. Iterative methods for linear systems 8. Finite Markov Chains
Abstract: 1. Matrices which leave a cone invariant 2. Nonnegative matrices 3. Semigroups of nonnegative matrices 4. Symmetric nonnegative matrices 5. Generalized inverse- Positivity 6. M-matrices 7. Iterative methods for linear systems 8. Finite Markov Chains 9. Input-output analysis in economics 10. The Linear complementarity problem 11. Supplement 1979-1993 References Index.
6,572 citations
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
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
TL;DR: It is shown that if G is a graph on n vertices, then E(G)@?n21+n must hold, and an infinite family of graphs for which this bound is sharp is given.
248 citations
TL;DR: In this paper, the spectral radius of the adjacency matrix and the Laplacian matrix of a simple undirected graph is analyzed in terms of the degrees and the 2-degrees of vertices.
161 citations