Some upper and lower bounds for $D_{\alpha}$-energy of graphs
DOI:
https://doi.org/10.13069/jacodesmath.v10i2.176Keywords:
Generalized distance matrix, Generalized distance energy, Distance (signless Laplacian) matrix, Transmission regular graph, Generalized distance spectral spreadAbstract
The generalized distance matrix of a connected graph $G$, denoted by $D_{\alpha}(G)$, is defined as
$D_{\alpha}(G)=\alpha Tr(G)+(1-\alpha)D(G), ~~~~ 0\leq \alpha\leq 1$.
Here, $D(G)$ is the distance matrix and $Tr(G)$ represents the vertex transmissions. Let $\partial_{1}\geq \partial_{2}\geq \cdots \geq \partial_{n}$ be the eigenvalues of $D_{\alpha}(G)$ and let $W(G)$ be the Wiener index. The generalized distance energy of $G$ can be defined as
$E^{D_{\alpha}}(G)=\displaystyle\sum_{i=1}^{n}\left|\partial_i-\frac{2\alpha W(G)}{n}\right|$.
In this paper, we develop some new theory regarding the generalized distance energy $E^{D_{\alpha}}(G)$ for a connected graph $G$. We obtain some sharp upper and lower bounds for $E^{D_{\alpha}}(G)$ connecting a wide range of parameters in graph theory including the maximum degree $\Delta$, the Wiener index $W(G)$, the diameter $d$, the transmission degrees, and the generalized distance spectral spread $D_{\alpha}S(G)$. We characterized the special graph classes that attain the bounds.
Received: 26 November 2020 Accepted: 31 December 2022Downloads
References
A. Alhevaz, M. Baghipur, K. C. Das, and Y. Shang, Sharp bounds on (generalized) distance energy of graphs, Mathematics 8(3) (2020) 426.
A. Alhevaz, M. Baghipur, H. A. Ganie, and Y. Shang, Bounds for the generalized distance eigenvalues of a graph, Symmetry 11(12) (2019) 1529.
A. Alhevaz, M. Baghipur, H. A. Ganie, and Y. Shang, On the generalized distance energy of graphs, Mathematics 8(1) (2020) 17.
A. Alhevaz, M. Baghipur, E. Hashemi, and H. S. Ramane, On the distance signless Laplacian spectrum of graphs, Bull. Malays. Math. Sci. Soc. 42(5) (2019) 2603–2621.
A. Alhevaz, M. Baghipur, and S. Paul, On the distance signless Laplacian spectral radius and the distance signless Laplacian energy of graphs, Discrete Math. Algorithms Appl. 10(3) (2018) 1850035.
M. Aouchiche and P. Hansen, Distance spectra of graphs: a survey, Linear Algebra Appl. 458 (2014) 301–386.
M. Aouchiche and P. Hansen, Two Laplacians for the distance matrix of a graph, Linear Algebra Appl. 439(1) (2013) 21–33.
F. Ashraf, G. R. Omidi, and B. Tayfeh-Rezaie, On the sum of signless Laplacian eigenvalues of a graph, Linear Algebra Appl. 438(11) (2013) 4539–4546.
S. Y. Cui, J. X. He, and G. X. Tian, The generalized distance matrix, Linear Algebra Appl. 563 (2019) 1–23.
R. C. Díaz and O. Rojo, Sharp upper bounds on the distance energies of a graph, Linear Algebra Appl. 545 (2018) 55–75.
H. A. Ganie, A. Alhevaz, and M. Baghipur, On the sum of the generalized distance eigenvalues of graphs, Discrete Math. Algorithms Appl. 13(1) (2021) 2050100.
H. A. Ganie and S. Pirzada, On the bounds for signless Laplacian energy of a graph, Discrete Appl. Math. 228 (2017) 3–13.
I. Gutman, The energy of a graph, Ber. Math.-Statist. Sekt. Forschungsz. Graz 103 (1978) 1–22.
I. Gutman, The energy of a graph: Old and new results, in: A. Betten, A. Kohnert, R. Laue, and A. Wassermann (Eds.), Algebraic Combinatorics and Applications, Springer, Berlin, Heidelberg, 2001, pp. 196–211.
G. Indulal, Sharp bounds on the distance spectral radius and the distance energy of graphs, Linear Algebra Appl. 430(1) (2009) 106–113.
G. Indulal, I. Gutman, and A. Vijayakumar, On distance energy of graphs, MATCH Commun. Math. Comput. Chem. 60(2) (2008) 461–472.
X. Li, Y. Shi, and I. Gutman, Graph Energy, Springer, New York, 2012.
H. Minc, Nonnegative Matrices, John Wiley and Sons Inc., New York, 1988.
D. S. Mitrinović, J. E. Pečarić, and A. M. Fink, Classical and New Inequalities in Analysis, Kluwer Academic Publishers, Dordrecht, 1993.
D. S. Mitrinović and P. M. Vasić, Analytic Inequalities, Springer-Verlag, Berlin, Heidelberg, New York, 1970.
S. Pirzada and H. A. Ganie, On the Laplacian eigenvalues of a graph and Laplacian energy, Linear Algebra Appl. 486 (2015) 454–468.
S. Pirzada, H. A. Ganie, A. Alhevaz, and M. Baghipur, On spectral spread of generalized distance matrix of a graph, Linear Multilinear Algebra 70(15) (2022) 2819–2835.
S. Pirzada, H. A. Ganie, B. A. Rather, and R. U. Shaban, On generalized distance energy of graphs, Linear Algebra Appl. 603 (2020) 1–19.
J. Yang, L. You, and I. Gutman, Bounds on the distance Laplacian energy of graphs, Kragujevac J. Math. 37(2) (2013) 245–255.
