The forgotten topological coindex (also called Lanzhou index) is defined for a simple connected graph G as the sum of the terms du2+dv2 over all non-adjacent vertex pairs uv of G, where du denotes the degree of the vertex u in G. In this paper, we present some inequalities for the forgotten topological coindex in terms of some graph parameters such as the order, size, number of pendent vertices, minimal and maximal vertex degrees, and minimal non-pendent vertex degree. We also study the relation between this invariant and some well-known graph invariants such as the Zagreb indices and coindices, multiplicative Zagreb indices and coindices, Zagreb eccentricity indices, eccentric connectivity index and coindex, and total eccentricity. Exact formulae for computing the forgotten topological coindex of double graphs and extended double cover of a given graph are also proposed.

In this paper, we explore several properties of Sombor coindex of a finite simple graph and we derive a bound for the total Sombor index. We also explore its relations to the Sombor index, the Zagreb coindices, forgotten coindex and other important graph parameters. We further compute the bounds of the Somber coindex of some graph operations and derived explicit formulae of Sombor coindex for some well-known graphs as application.

Let $G=(V,E)$, $V=\left\{ v_{1},v_{2},\ldots ,v_{n}\right\}$, be a simple graph of order $n$ and size $m$, without isolated vertices. The Sombor coindex of a graph $G$ is defined as  $\overline{SO}(G)=\sum_{i\nsim j}\sqrt{d_i^2+d_j^2}$ , where $d_i= d(v_i)$ is a degree of vertex $v_i$, $i=1,2,\ldots , n$. In this paper we investigate a relationship  between  Sombor coindex and a number of other topological coindices.

In the last years, Naji et al. have introduced leap Zagreb indices conceived depending on the second degrees of vertices, where the second degree of a vertex $v$ in a graph $G$ is equal to the number of its second neighbors and denoted by $d_2(v/G)$.  Analogously, the leap Zagreb coindices were introduced by Ferdose and Shivashankara. The first leap Zagreb coindex of a graph is defined as  $\overline{L_1}(G)=\sum_{uv\not\in E_2(G)}(d_2(u)+d_2(v))$, where $E_2(G)$ is the 2-distance (second) edge set of $G$, In this paper, we present explicit exact expressions for the first leap Zagreb coindex $\overline{L_1}(G)$ of some graph operations.

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The second Zagreb indexM2(G) is equal to the sum of the products of the degrees of pairs of adjacent vertices and the second Zagreb coindex M2(G) is equal to the sum of the products of the degrees of pairs of non-adjacent vertices. Kovijanić Vukić ević and Popivoda (Iranian J. Math. Chem. 5 (2014) 19– 29) prove that for any chemical tree of order n  5, M2(T )   8n − 26 n  0, 1 (mod 3) 8n − 24 otherwise. In this paper we present a generalization of the aforementioned bound for all trees in terms of the order and maximum degree. We also give a lower bound on the second Zagreb coindex of trees.

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Let G = (V; E), V = fv1; v2; : : :; vng, be a simple graph with n vertices, m edges and a sequence of vertex degrees
= d1  d2 


 dn =  , di = d(vi). If vertices vi and vj are adjacent in G, it is denoted as i  j, otherwise, we write i  j. The  rst Zagreb index is vertex-degree-based graph invariant de ned as M1(G) = Pn i=1, whereas the  rst Zagreb coindex is de ned as M1(G) = P d 2 i i  j (di + dj ). A couple of new upper and lower bounds for M1(G), as well as a new upper bound for M1(G), are obtained.

The forgotten topological index of a molecular graph G is defined as F (G) = S nÎv (G) d (v)3, where d (v) denotes the degree of vertex v in G. the first through the sixth smallest forgotten indices among all trees, the first through the third smallest forgotten indices among all connected graph with cyclomatic number g = 1, 2, the first through the fourth for g = 3, and the first and the second for g = 4, 5 are determined. These results are compared with those obtained for the first Zagreb index.