University of Isfahan Transactions on Combinatorics 2251-8657 3 3 2014 09 01 General Randic matrix and general Randic energy 21 33 5451 10.22108/toc.2014.5451 EN Ran Gu Center for Combinatorics, nankai University, Tianjin, China Fei Huang Center for Combinatorics, Nankai University, Tianjin, China Xueliang Li Center for Combinatorics, Nankai University, Tianjin 300071, China Journal Article 2014 05 12 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 $\alpha=-1/2$‎. https://toc.ui.ac.ir/article_5451_12d35e7ada9a784103a3b287d094dee6.pdf