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 $v_i$, $i = 1, 2,cdots, n$. inspired by the randi'c matrix and the general randi'cindex of a graph, we introduce the concept of general randi'cmatrix $textbf{r}_alpha$ of $g$, which is defined by$(textbf{r}_alpha)_{i,j}=(d_id_j)^alpha$ if $v_i$ and $v_j$ areadjacent, and zero otherwise. s...

<abstract><p>The zeroth-order general Randić index of graph $ G = (V_G, E_G) $, denoted by ^0R_{\alpha}(G) is the sum items (d_{v})^{\alpha} over all vertices v\in V_G where \alpha a pertinently chosen real number. In this paper, we obtain sharp upper and lower bounds on ^0R_{\alpha} trees with given domination number \gamma for \alpha\in(-\infty, 0)\cup(1, \infty) \alpha\in(0, 1) r...

In this paper, we show that in the class of connected graphs G of order n ≥ 3 having girth at least equal to k, 3 ≤ k ≤ n, the unique graph G having minimum general sum-connectivity index χα(G) consists of Ck and n−k pendant vertices adjacent to a unique vertex of Ck, if−1 ≤ α < 0. This property does not hold for zeroth-order general Randić index Rα(G).

The general Randić index Rα(G) of a (chemical) graph G, is defined as the sum of the weights (d(u)d(v))α of all edges uv of G, where d(u) denotes the degree of a vertex u in G and α an arbitrary real number, which is called the Randić index or connectivity index (or branching index) for α = −1/2 proposed by Milan Randić in 1975. The paper outlines the results known for the (general) Randić inde...