On the vertex-degree based invariants of digraphs

Let $D=(V,A)$ be a digraphs without isolated vertices. A vertex-degree based invariant $I(D)$ related to a real function $\varphi$ of $D$ is defined as a summation over all arcs, $I(D) = \frac{1}{2}\sum_{uv\in A}{\varphi(d_u^+,d_v^-)}$, where $d_u^+$ (resp. $d_u^-$) denotes the out-degree (resp. in-degree) of a vertex $u$. In this paper, we give the extremal values and extremal digraphs of $I(D)$ over all digraphs with $n$ non-isolated vertices. Applying these results, we obtain the extremal values of some vertex-degree based topological indices of digraphs, such as the Randi\'{c} index, the Zagreb index, the sum-connectivity index, the $GA$ index, the $ABC$ index and the harmonic index, and the corresponding extremal digraphs.


Introduction
A digraph D = (V, A) is an ordered pair (V, A) consisting of a non-empty finite set V of vertices and a finite set A of ordered pairs of distinct vertices called arcs (in particular, D has no loops). If a ∈ A is an arc from vertex u to vertex v, then we indicate this by writing a = uv. The vertex u is the tail of a, and the vertex v its head. The out-degree (resp. in-degree) of a vertex u, denoted by d + u (resp. d − u ) is the number of arcs with tail u (resp. with head u). A vertex u for which d + u = d − u = 0 is called an isolated vertex. We denote by D n the set of all digraphs with n non-isolated vertices.
Recently, J. Monsalve and J. Rada [7] extended the concept of vertex-degree based topological indices of graphs to digraphs. They obtained the extremal values of the Randić index of digraphs over D n , and found the extremal values of the Randić index over the set of all oriented trees with n vertices. Also, they found the extremal values of the Randić index over the set of all orientations of the path, the cycle with n vertices and the hypercube H d of dimension d, respectively.
All the digraphs considered in this paper are strict, i.e., no loops and no two arcs with the same ends have the same orientation.
A vertex-degree-based (VDB, for short) VDB invariant (or VDB topological index) I(D) related to a real function φ of a digraph D with n non-isolated vertices is defined as where φ ij = φ(i, j) and a ij is the number of arcs in D of the form uv such that d + u = i and d − v = j, i.e., (i, j)-arcs in D.
Recall that if G is a graph, we can identify G with the symmetric digraph − → G by replacing every edge of G with a pair of symmetric arcs. Under this correspondence, for any VDB topological index φ with φ ij = φ ji (symmetric) and m ij the number of edges in G joining vertices of degree i and j. In other words, The VDB topological index of digraphs is a generalization of the concept of VDB topological index of graphs.
In fact, a VDB topological index I(D) of a digraph is an invariant based on the weights of all arcs depends on the out degrees of their tails and the in-degrees of their heads, i.e., where φ(x, y) is a real function of x and y with φ(x, y) ≥ 0 and φ(x, y) = φ(y, x). Zagreb index for α = − 1 2 , α = 1 and α = −1, respectively. For these indices of graphs, see [1,6,8,9]. (ii) If φ(x, y) = (x + y) α , then I(D) is the general sum-connectivity index of a digraph D. Further, I(G) is the sum-connectivity index and the first Zagreb index for α = − 1 2 and α = 1, respectively. See [5,8,11,12] for graphs.
(iii) If φ(x, y) = √ xy 1 2 (x+y) , then I(D) is the first geometric-arithmetic index GA of a digraph D. See [10] for the first geometric-arithmetic index of a graph.
(iv) If φ(x, y) = √ x+y−2 xy , then I(G) is the atom-bond connectivity (ABC) index of a digraph D. See [3] for the atom-bond connectivity index of a graph.
(v) If φ(x, y) = 2 x+y , then I(D) is the harmonic index of a digraph D. See [4] for the harmonic index of a graph.
In this paper, we give the extremal values and extremal graphs of the VDB topological indices over all digraphs with n non-isolated vertices by a unified linear-programming modeling, and provide a unified approach to determining some extremal values and characterizing extremal diraphs of Randić index, Zagreb index, sum-connectivity index, GA index, ABC index and harmonic index by using the linear programming methods.

General results on VDB invariants
Let D be a digraph on n ≥ 2 vertices without isolated vertices and a ij the number of arcs of D from vertices of out-degree i to vertices of in-degree j. If φ is symmetric, i.e. φ ij = φ ji for all 1 ≤ i < j ≤ n−1, then we can simplify the expression in (1) in the following where p ij = a ij + a ji for i ̸ = j and 1 ≤ i, j ≤ n − 1, and p ii = a ii for all i = 1, 2, · · · , n − 1.
Note that p ij = p ji for all 1 ≤ i, j ≤ n − 1, and where n i is the number of vertices of D with out-degree i or in-degree i. Also, The digraphs with n non-isolated vertices which satisfy the following conditions are of great interest to us i.e., a digraph with only (1, n − 1)-or (n − 1, 1)-arcs and the out-degree or in-degree of each vertex greater than 0. The digraph obtained from the star on n vertices by replacing each of its edges with a pair of symmetric arcs satisfies (5). The converse of this example does not hold since i.e., the digraphs with only (i, i)-arcs (1 ≤ i ≤ n − 1) and the out-degree or in-degree of each vertex equal i.e., the digraphs with only (i, i)-arcs (1 ≤ i ≤ n−1) and the out-degree or in-degree of each vertex greater than 0. The dircted cycle − → C n on n vertices satisfies (7). All digraphs with n non-isolated vertices in which each component is regular satisfy (7). The converse of this example does not hold since We try to find min(I(G)) and max(I(G)) under the constraints (3) and (4)

Applications
In this section, we give some results on Randić index, Zagreb index, sum-connectivity index, GA index and ABC index of digraphs by using Theorems 1-3.

The general Randić index of digraphs
with n non-isolated vertices. In particular, R α (D) is the Randić index, the second Zagreb index and the modified Zagreb index of a digraph for α = − 1 2 , α = 1 and α = −1, respectively.
and φ ij ≤ n−1 2 ( 1 i + 1 j )φ n−1,n−1 with equality if and only if (a) i = j = n − 1 for − 1 2 < α < +∞, or (b) i = j for α = − 1 2 . By Theorems 2(ii) and 3(ii), we have with equality if and only if (a) D is the digraph obtained from K n by replacing each edge with a pair of symmetric arcs for − 1 2 < α < +∞, or (b) D is a digraph satisfied (7). So, (a) the digraph with the maximal general Randić index (including the second Zagreb index) for − 1 2 < α < +∞ is the digraph obtained from K n by replacing each edge with a pair of symmetric arcs; (b) the digraphs with the maximal Randić index are those satisfied (7), see Theorem 3.7 in [7].  (7).
By Theorem 1(i), we have with equality if and only if D is the digraph − → K 1,n−1 or − → K n−1,1 for sufficiently large n.

The general sum-connectivity index of digraphs
By Theorems 2(ii) and 3(ii), we have with equality if and only if (a) D is the digraph obtained from the complete graph K n by replacing each edge with a pair of symmetric arcs, or (b) D satisfies (7).
Especially, this shows that the graph with the maximal sum-connectivity index, or the maximal first Zagreb index is K n among all graphs of order n.

Corollary 7.
If D ∈ D n , then (a) χ α (D) ≤ 2 α−1 n(n − 1) α+1 for − 1 2 < α < +∞ with equality if and only if D is the digraph obtained from K n by replacing each edge with a pair of symmetric arcs; (b) with equality if and only if D satisfies (7).

The atom-bond connectivity index of digraphs
is the ABC index of a digraph D. This shows that the digraphs with the maximal ABC index over D n is the digraph obtained from K n by replacing each edge with a pair of symmetric arcs.  with equality if and only if D is a digraph satisfied (7).