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Spectral properties of geometric–arithmetic index

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  • Rodríguez, José M.
  • Sigarreta, José M.

Abstract

The concept of geometric–arithmetic index was introduced in the chemical graph theory recently, but it has shown to be useful. One of the main aims of algebraic graph theory is to determine how, or whether, properties of graphs are reflected in the algebraic properties of some matrices. The aim of this paper is to study the geometric–arithmetic index GA1 from an algebraic viewpoint. Since this index is related to the degree of the vertices of the graph, our main tool will be an appropriate matrix that is a modification of the classical adjacency matrix involving the degrees of the vertices. Moreover, using this matrix, we define a GA Laplacian matrix which determines the geometric–arithmetic index of a graph and satisfies properties similar to the ones of the classical Laplacian matrix.

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  • Rodríguez, José M. & Sigarreta, José M., 2016. "Spectral properties of geometric–arithmetic index," Applied Mathematics and Computation, Elsevier, vol. 277(C), pages 142-153.
    • Handle: RePEc:eee:apmaco:v:277:y:2016:i:c:p:142-153
    DOI: 10.1016/j.amc.2015.12.046
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    References listed on IDEAS

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    1. Das, Kinkar Ch. & Mojallal, Seyed Ahmad, 2016. "Extremal Laplacian energy of threshold graphs," Applied Mathematics and Computation, Elsevier, vol. 273(C), pages 267-280.
    2. Shi, Yongtang, 2015. "Note on two generalizations of the Randić index," Applied Mathematics and Computation, Elsevier, vol. 265(C), pages 1019-1025.
    3. Renqian, Suonan & Ge, Yunpeng & Huo, Bofeng & Ji, Shengjin & Diao, Qiangqiang, 2015. "On the tree with diameter 4 and maximal energy," Applied Mathematics and Computation, Elsevier, vol. 268(C), pages 364-374.
    4. Das, Kinkar Ch. & Mojallal, Seyed Ahmad & Gutman, Ivan, 2016. "On energy and Laplacian energy of bipartite graphs," Applied Mathematics and Computation, Elsevier, vol. 273(C), pages 759-766.
    5. Li, Xueliang & Qin, Zhongmei & Wei, Meiqin & Gutman, Ivan & Dehmer, Matthias, 2015. "Novel inequalities for generalized graph entropies – Graph energies and topological indices," Applied Mathematics and Computation, Elsevier, vol. 259(C), pages 470-479.
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    Cited by:

    1. Vujošević, Saša & Popivoda, Goran & Kovijanić Vukićević, Žana & Furtula, Boris & Škrekovski, Riste, 2021. "Arithmetic–geometric index and its relations with geometric–arithmetic index," Applied Mathematics and Computation, Elsevier, vol. 391(C).
    2. Shao, Yanling & Gao, Yubin, 2019. "The maximal geometric-arithmetic energy of trees with at most two branched vertices," Applied Mathematics and Computation, Elsevier, vol. 362(C), pages 1-1.
    3. Cui, Qing & Zhong, Lingping, 2017. "The general Randić index of trees with given number of pendent vertices," Applied Mathematics and Computation, Elsevier, vol. 302(C), pages 111-121.
    4. Liu, Chang & Pan, Yingui & Li, Jianping, 2021. "On the geometric-arithmetic Estrada index of graphs," Applied Mathematics and Computation, Elsevier, vol. 391(C).
    5. Milovanović, E.I. & Milovanović, I.Ž. & Matejić, M.M., 2018. "Remark on spectral study of the geometric–arithmetic index and some generalizations," Applied Mathematics and Computation, Elsevier, vol. 334(C), pages 206-213.
    6. Mahdi Sohrabi-Haghighat & Mohammadreza Rostami, 2017. "The minimum value of geometric-arithmetic index of graphs with minimum degree 2," Journal of Combinatorial Optimization, Springer, vol. 34(1), pages 218-232, July.

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