General Randic matrix and general Randic energy

Document Type : Research Paper


1 Center for Combinatorics, nankai University, Tianjin, China

2 Center for Combinatorics, Nankai University, Tianjin, China

3 Center for Combinatorics, Nankai University, Tianjin 300071, China


Let $G$ be a simple graph with vertex set $V(G) = \{v_1‎, ‎v_2,\ldots‎, ‎v_n\}$ and $d_i$ the degree of its vertex $v_i$‎, ‎$i = 1‎, ‎2‎, ‎\dots‎, ‎n$‎. ‎Inspired by the Randic matrix and the general Randic‎ ‎index of a graph‎, ‎we introduce the concept of general Randi'c‎ ‎matrix $\textbf{R}_\alpha$ of $G$‎, ‎which is defined by‎ $(\textbf{R}_\alpha)_{i,j}=(d_id_j)^\alpha$ if $v_i$ and $v_j$ are‎ ‎adjacent‎, ‎and zero otherwise‎. ‎Similarly‎, ‎the general Randic‎ ‎eigenvalues are the eigenvalues of the general Randic} matrix‎, ‎the greatest general Randic eigenvalue is the general Randic‎ ‎spectral radius of $G$‎, ‎and the general Randic energy is the sum‎ ‎of the absolute values of the general Randic eigenvalues‎. ‎In ‎this paper‎, ‎we prove some properties of the general Randi'c matrix‎ ‎and obtain lower and upper bounds for general Randic energy‎, ‎also‎, ‎we get some lower bounds for general Randic spectral‎ ‎radius of a connected graph‎. ‎Moreover‎, ‎we give a new sharp upper‎ ‎bound for the general Randic energy when $\alpha=-1/2$‎.


Main Subjects

B. Bollob\'{a}s and P. Erd\H{o}s (1998). Graphs of extremal weights. Ars Combin.. 50, 225-233 J. A. Bondy and U. S. R. Murty (2008). Graph Theory. Springer New York. 244 \c{S}. B. Bozkurt, A. D. G\"{u}ng\"{o}r and I. Gutman (2010). Randi\'{c} spectral radius and Randi\'{c} energy. MATCH Commun. Math. Comput. Chem.. 64, 321-334 S. B. Bozkurt, A. D. Gongor, I. Gutman and A. S. Cevik (2010). Randi\'c Matrix and Randi\'c Energy. MATCH Commun. Math. Comput. Chem.. 64, 239-250 D. Cvetkovic, M. Doob and H. Sachs (1980). Spectra of Graphs--Theory and Application. Academic Press, New York. 87 M. Cavers, S. Fallat and S. Kirkland (2010). On the normalized Laplacian energy and general Randi\'{c} index $R_{-1}$ of graphs. Linear Algebra Appl.. 433, 172-190 B. Furtula and I. Gutman (2013). Comparing energy and Randi\'c energy. Macedonian Journal of Chemistry and Chemical Engineering. 32 (1), 117-123 F. R. Gantmacher (1959). The Theory of Matrices. Chelsea, New York. I. Gutman, B. Furtula and \c{S}. B. Bozkurt (2014). On Randi\' c energy. Linear Algebra Appl.. 442, 50-57 C. Godsil and G. Royle (2001). Algebraic Graph Theory. Springer, New York. M. Randi\'{c} (1975). On characterization of molecular branching. J. Amer. Chem. Soc.. 97, 6609-6615 L. Shi (2009). Bounds on Randi\'{c} indices. Discrete Math.. 309 (16), 5238-5241