Some remarks on the sum of the inverse values of the normalized signless Laplacian eigenvalues of graphs

Milovanovic, Igor; Milovanovic, Emina; Matejic, Marjan; Bozkurt Altındağ, Serife Burcu

doi:10.22049/cco.2021.26987.1173

تعداد نشریات	6
تعداد شماره‌ها	123
تعداد مقالات	1,565
تعداد مشاهده مقاله	1,687,934
تعداد دریافت فایل اصل مقاله	1,573,248

	Some remarks on the sum of the inverse values of the normalized signless Laplacian eigenvalues of graphs
Communications in Combinatorics and Optimization
دوره 6، شماره 2، اسفند 2021، صفحه 259-271 اصل مقاله (411.56 K)
نوع مقاله: Original paper
شناسه دیجیتال (DOI): 10.22049/cco.2021.26987.1173
نویسندگان
Igor Milovanovic^* ¹؛ Emina Milovanovic²؛ Marjan Matejic²؛ Serife Burcu Bozkurt Altındağ³
¹Faculty of Electronic Engineering, Nis, Serbia
²Faculty of Electronic Engineering
³Yenikent Kardelen Konutlari, Selcuklu
چکیده
Let $G=(V,E)$, $V=\{v_1,v_2,\ldots,v_n\}$, be a simple connected graph with $n$ vertices, $m$ edges and a sequence of vertex degrees $d_1\geq d_2\geq\cdots\geq d_n>0$, $d_i=d(v_i)$. Let ${A}=(a_{ij})_{n\times n}$ and ${D}=\mathrm{diag}d_1,d_2,\ldots , d_n)$ be the adjacency and the diagonal degree matrix of $G$, respectively. Denote by ${\mathcal{L}^+}(G)={D}^{-1/2} (D+A) {D}^{-1/2}$ the normalized signless Laplacian matrix of graph $G$. The eigenvalues of matrix $\mathcal{L}^{+}(G)$, $2=\gamma _{1}^{+}\geq \gamma_{2}^{+}\geq \cdots \geq \gamma_{n}^{+}\geq 0$, are normalized signless Laplacian eigenvalues of $G$. In this paper some bounds for the sum $K^{+}(G)=\sum_{i=1}^n\frac{1}{\gamma _{i}^{+}}$ are considered.
کلیدواژه‌ها
Laplacian matrix؛ normalized signless Laplacian matrix؛ eigenvalues

مراجع
[1] Ş.B. Bozkurt Altındağ, Note on the sum of powers of normalized signless Laplacian eigenvalues of graphs, Math. Interdisc. Res. 4 (2019), no. 2, 171–182. [2] Ş.B. Bozkurt Altındağ, Sum of powers of normalized signless Laplacian eigenvalues and Randić (normalized) incidence energy of graphs, Bull. Inter. Math. Virtual Inst. 11 (2021), no. 1, 135–146. [3] Ş.B. Bozkurt Altındağ, A.D. Güngör, I. Gutman, and A.S. Çevik, Randić matrix and Randić energy, MATCH Commun. Math. Comput. Chem. 64 (2010), no. 1, 239–250. [4] S. Butler, Algebraic aspects of the normalized Laplacian, in: Recent Trends in Combinatorics, IMA Vol. Math. Appl. 159, Springer, 2016, pp. 295–315. [5] M. Cavers, S. Fallat, and S. Kirkland, On the normalized Laplacian energy and general Randi´c index R−1 of graphs, Lin. Algebra Appl. 433 (2010), no. 1, 172–190. [6] B. Cheng and B. Liu, The normalized incidence energy of a graph, Lin. Algebra Appl. 438 (2013), no. 11, 4510–4519. [7] F.R.K. Chung, Spectral Graph Theory, no. 92, Amer. Math. Soc. Providence, 1997. 8] D. Cvetković, M. Doob, and H. Sachs, Spectra of Graphs, Academic press, New York, 1980. [9] K.C. Das, A.D. Gungor, and Ş.B. Bozkurt Altındağ, On the normalized Laplacian eigenvalues of graphs, Ars Combin. 118 (2015), 143–154. [10] Z. Du, A. Jahanbani, and S.M. Sheikholeslami, Relationships between Randi´c index and other topological indices, Commun. Comb. Optim. 6 (2021), no. 1, 137–154. [11] S. Furuichi, On refined Young inequalities and reverse inequalities, J. Math. Inequal. 5 (2011), 21–31. [12] R. Gu, F. Huang, and X. Li, Randić incidence energy of graphs, Trans. Comb. 3 (2014), no. 4, 1–9. [13] J.G. Kemeny and J.L. Snel, Finite Markov Chains, Van Nostrand, Princeton, N.J., 1960. [14] M. Levene and G. Loizou, Kemeny’s constant and the random surfer, Amer. Math. Monthly 109 (2002), no. 8, 741–745. [15] J. Li, J.-M. Guo, and W.C. Shiu, Bounds on normalized Laplacian eigenvalues of graphs, J. Inequal. Appl. 2014 (2014), no. 1, ID: 316. [16] M. Matejić, I. Milovanović, and E. Milovanović, Remarks on the degree Kirchoff index, Kragujevac J. Math. 43 (2019), no. 1, 15–21. [17] E.I. Milovanović, M.M. Matejić, and I.Z. Milovanović, On the normalized Laplacian spectral radius, Laplacian incidence energy and Kemeny’s constant, Lin. Algebra Appl. 582 (2019), 181–196. [18] D.S. Mitrinović and P.M. Vasić, Analytic Inequalities, Springer Verlag, BerlinHeidelberg-New York, 1970. [19] M. Randić, Characterization of molecular branching, J. Am. Chem. Soc. 97 (1975), no. 23, 6609–6615. [20] S. Sun and K.C. Das, On the second largest normalized Laplacian eigenvalue of graphs, Appl. Math. Comput. 348 (2019), 531–541. [21] B. Zhou and N. Trinajstić, On resistance-distance and Kirchhoff index, J. Math. Chem. 46 (2009), no. 1, 283–289. [22] P. Zumstein, Comparison of spectral methods through the adjacency matrix and the Laplacian of a graph, Th. Diploma, ETH Z¨urich, 2005.
آمار تعداد مشاهده مقاله: 931 تعداد دریافت فایل اصل مقاله: 1,454

سامانه مدیریت نشریات علمی. قدرت گرفته از سیناوب

پیوندهای مفید

آمار

Some remarks on the sum of the inverse values of the normalized signless Laplacian eigenvalues of graphs