Randi{\'c} INDEX belongs to the most well-known topological indices in chemical graph theory. In this paper, we find upper bound for the Randi{\'c} INDEX of trees in terms of the order and the total domination number. The extremal trees are characterized.

Let G be a simple graph with vertex set V (G) = {v1, v2,…, vn} and di the degree of its vertex vi, i=1, 2,…, n. Inspired by the RANDIC matrix and the general RANDIC INDEX of a graph, we introduce the concept of general RANDIC matrix Ra of G, which is defined by (Ra) i, j= (didj) a if vi and vj 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 RANDIC 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 a= -1/2.

Let G be a simple graph with an orientation s, which assigns to each edge a direction so that Gs becomes a directed graph. G is said to be the underlying graph of the directed graph Gs. In this paper, we define a weighted skew adjacency matrix with Randc weight, the skew Randic matrix RS (Gs), of Gs as the real skew symmetric matrix [(rs)ij] where (rs)ij = (didj) -1/2 and (rs)ji = -(didj) -1/2 if vi ® vj is an arc of Gs, otherwise (rs)ij = (rs)ji = 0. We derive some properties of the skew RANDIC energy of an oriented graph. Most properties are similar to those for the skew energy of oriented graphs. But, surprisingly, the extremal oriented graphs with maximum or minimum skew RANDIC energy are completely different, no longer being some kinds of oriented regular graphs.

A topological INDEX is a numerical quantity for the structure graph of the molecule and it can be represented through graph theory. Also, its application not only in the field of chemistry can also be applied in areas including computer science, networking, etc. Hence, this paper introduces the RANDIC INDEX of a vague graph and vague subgraph with their properties. The upper and lower boundaries of the RANDIC INDEX of vague graphs are studied with some isomorphic properties. Likewise, the RANDIC INDEX of directed vague graphs is introduced. Finally, an application of the RANDIC INDEX in construction has been presented.

Let G be a simple graph with vertex set V (G) ={v1, v2,… vn} and edge set E (G) = {e1, e2,… em}. Similar to the RANDIC matrix, here we introduce the RANDIC incidence matrix of a graph G, denoted by IR (G), which is defined as the n×m matrix whose (I, j) -entry is (di) -1/2 if vi is incident to ej and 0 otherwise. Naturally, the RANDIC incidence energy IRE of G is the sum of the singular values of IR (G). We establish lower and upper bounds for the RANDIC incidence energy. Graphs for which these bounds are best possible are characterized. Moreover, we investigate the relation between the RANDIC incidence energy of a graph and that of its subgraphs. Also we give a sharp upper bound for the RANDIC incidence energy of a bipartite graph and determine the trees with the maximum RANDIC incidence energy among all n-vertex trees. As a result, some results are very different from those for incidence energy.