On the transmission-based graph topological indices

The distance $d(u,v)$ between the vertices $u$ and $v$ of a connected graph $G$ is defined as the number of edges in a minimal path connecting them. The \emph{transmission} of a vertex $v$ of $G$ is defined by $\sigma(v)=\sum\limits_{u\in V(G)}{d(v,u)}$. In this article we aim to define some transmission-based topological indices. We obtain lower and upper bounds on these indices and characterize graphs for which these bounds are best possible. Finally, we find these indices for various graphs using the group of automorphisms of $G$. This is an efficient method of finding these indices especially when the automorphism group of $G$ has a few orbits on $V(G)$ or $E(G)$.


Introduction and Preliminaries
Using the standard terminology in graph theory, we refer the reader to [42]. The degree d(u) of the vertex u ∈ V (G) is the number of the edges incident to u. The edge of the graph G connecting the vertices u and v is denoted by uv.
The role of molecular descriptors (especially topological descriptors) is remarkable in mathematical chemistry especially in QSPR/QSAR investigations. In mathematical chemistry, the first Zagreb index M 1 (G) and the second Zagreb index M 2 (G) belong to the family of the most important degree-based molecular descriptors. They are defined as [22], [23], [25], [31], [36] Similarly, the first variable Zagreb index and the second variable Zagreb index are defined as [33], [36], [44] M λ where λ is a real number.
The Randic index R(G), the ordinary sum-connectivity index X(G), the harmonic index H(G) and geometric-arithmetic index GA(G) are also widely used degree-based topological indices [39], [48], [17], [43], [46], [47]. By definition, Let ∆ = ∆(G) and δ = δ(G) be the maximum and the minimum degrees, respectively, of vertices of G. The average degree of G is 2m n . A connected graph G is said to be bidegreed 2 with degrees ∆ and δ ( ∆ > δ ≥ 1), if at least one vertex of G has degree ∆ and at least one vertex has degree δ, and if no vertex of G has degree different from ∆ or δ. A connected bidegreed bipartite graph is called semi-regular if each vertex in the same part of a bipartition has the same degree. A graph G is called regular if all its vertices have the same degree, otherwise it is said to be irregular. In many applications and problems in theoretical chemistry, it is important to know how a given graph is irregular. The (vertex) regularity of a graph is defined in several approaches. Two most frequently used graph topological indices that measure how irregular a graph is, are the irregularity and variance of degrees. Let imb(e) = |d(u) − d(v)| be the imbalance of an edge e = uv ∈ E(G). In [1], the irregularity of G, which is a measure of irregularity of graph G, defined as The variance of degrees of graph G is defined as [7] Var(G) = 1 n Another measure of irregularity, which is called degree deviation, defined as [37] It is worth mentioning that s(G) n is noting but the mean deviation of the data set {d(u) | u ∈ V (G)}. The distance between the vertices u and v in graph G is denoted by d(u, v) and it is defined as the number of edges in a minimal path connecting them. The eccentricity ε(v) of a vertex v is the maximum distance from v to any other vertex. The diameter diam(G) of G is the maximum eccentricity among the vertices of G. The transmission (or status) of a vertex v of G is defined as u). A graph G is said to be transmission regular [3] if σ(u) = σ(v) for any vertex u and v of G. A transmission regular graph G is called k-transmission regular if there exists a positive integer k, for which σ(v) = k for any vertex v of G. In K n , the complete graph of order n, each vertex has transmission n − 1.
So it is (n − 1)-transmission regular. The the cycle C n and the complete bipartite graph K a,a are transmission regular. It has been verified that there exist regular and non-regular transmission regular graphs [3]. Consider the polyhedron depicted in Figure 1  The graph G RD of the rhombic dodecahedron is a bidegreed, semi-regular 28-transmission regular graph (See Figure 2). An interesting observation is that the 14-vertex polyhedral graph G RD depicted in Figure 2 is identical to the semi-regular graph published earlier in an alternative form in [3]. It is conjectured that G RD is the smallest non-regular, bipartite, polyhedral (3-connected) and transmission regular graph. If ω is a vertex weight of graph G, then one can see that It is easy to construct various transmission-based indices having the same structure as the known degree-based topological indices. Based on this analogy-concept, the corresponding transmission-based indices are defined.
Let us define the transmission Randić index RS(G), the transmission ordinary sumconnectivity index XS(G), the transmission harmonic index HS(G) and the transmission geometric-arithmetic index GAS(G) as follows: It follows that GAS(G) ≤ n 2 , with equality if and only if G is a transmission regular graph. The Wiener index W (G), the Balaban index J(G) and the sum-Balaban index SJ(G) represent a particular class of transmission-based topological indices. They are defined as [4], [5], [6], [10], [9], [49], [16], [21] In [40] the first transmission Zagreb index M S 1 (G) and the second transmission Zagreb index It is important to note that M S 1 (G) coincides with the degree distance DD(G) that was introduced in [11], [24] and [41] In fact by Eq. (3), Consequently, if G is a k-transmission regular graph with m vertices, then DD(G) = Let us propose the variable degree transmission Zagreb index M SD λ (G) and the variable transmission Zagreb index M S λ (G) as follows where λ is a real number.
The eccentric distance sum of a graph G, denoted by ξ d (G), defined as [20] It follows from Eq. (3) that Inspired from Eq. (6) and Eq. (7) we define two transmission-based irregularity as follows: Let G be a connected graph with n vertices and m edges. The transmission imbalance of an Let us define the transmission irregularity irr Tr (G) and the transmission variance Var Tr (G) of G as follows: irr Var Note that 2W (G) n is nothing but the vertex transmission average of graph G. It is obvious that Var Tr (G) is equal to zero if and only if G is transmission regular.
Let us also define the transmission-based topological indices QS e (G) and QS v,e (G) as Remark 1. Let G be an n-vertex graph. Comparing topological indices GAS(G) and QS v,e (G), we get Equalities hold in both sides simultaneously if and only if G is transmission regular.
2 Establishing lower and upper bounds Lemma 1. Let G be a connected graph with n ≥ 2 vertices and m edges. Then The equality on the right-hand sides holds if and only if G is isomorphic to S n . The equality on the left-hand sides holds if and only if G is transmission regular.
Proof. For an arbitrary edge uv of G, we have |σ(u) − σ(v)| ≤ n − 2. Therefore, It is trivial that in both formulas the equality on the right-hand side holds if and only if G isomorphic to S n , since the star is the only graph where equality holds for each edge.
Corollary 1. Let T be a tree with n ≥ 2 vertices. Then The equality on the right-hand sides holds if and only if G is isomorphic to S n . The equality on the left-hand sides holds if and only if G is transmission regular.
Proof. It is a consequence of Lemma 1 and the fact that a tree with n vertices has exactly n − 1 edges.
Corollary 2. Let G be a connected graph with n ≥ 2 vertices. Then The upper bounds are achieved if and only if G is isomorphic to S n and the lower bounds are achieved if and only if G is transmission regular.
Proof. It is a direct consequence of Lemma 1.

Lemma 2.
Let G be a connected graph with n ≥ 3 vertices and with maximum vertex degree ∆. Then for each arbitrary vertex u of G Remark 2. There are several graphs containing a vertex u for which σ(u) = n − 1. For example, σ(u) = d(u) = n − 1 for any vertex u of a complete graph K n .
Remark 3. Let G be a connected graph. It is easy to see that for any u ∈ V (G), σ(u) (ii) Let G be a connected graph with diam(G) ≤ 2. Then G is transmission regular if and only if G is regular.
with equality if and only if diam(G) ≤ 2.
Proof. It follows from Lemma 2 that and by Remark 2, the equality holds if and only if diam(G) ≤ 2.
Proposition 2. Let G be a connected graph with n vertices. Then with equality if and only if diam(G) ≤ 2.
Proof. It follows from Lemma 2 that It follows from Remark 3 that the equality holds if and only if diam(G) ≤ 2.
Lemma 3. Let G be a connected graph with n vertices and m edges. If diam(G) ≤ 2, then In particular, in both cases equality holds if and only if G is regular.
Proof. (i) It is a direct consequence of Lemma 1 and Remark 3. (ii) It follows directly from part (i).
Corollary 3. Let K p,q be the complete bipartite graph with p + q vertices and with parts of size p and q. Then In particular, the equalities in (i) and (ii) hold if and only if p = q.
Proof. (i) Since diam(K p,q ) = 2 and |E(K p,q )| = pq, it follows from Lemma 3 (i) that Specially, let n ≥ 2 and p = 1 and q = n − 1. Then K p,q is isomorphic to the star S n , (n = p + q). Consequently, we obtain that It follows from Lemma 3 that the equalities in (i)  Proposition 3. For the windmill graph W d(n) we have Proof. (i) Let E 0 be the set of strong edges of W d(n). It is easy to see that Since diam(W d(n)) = 2, it follows from Lemma 3 (i) that (ii) It follows from part (i) that .

Lemma 4 ( [32]
). Let P n be a path of order n, and let V ( The following is a direct consequence of Lemma 4.
Proposition 4. The transmission irregularity index of P n is given by , if n is even, , if n is odd.
For an edge uv of a connected graph G, define the positive integers N u and N v where N u is the number of vertices of G whose distance to vertex u is smaller than distance to vertex v, and analogously, N v is the number of vertices of G whose distance to the vertex v 13 is smaller than to u. The number of vertices equidistant from u and v is denoted by N uv .
An edge uv of G is called a distance-balanced edge if N u = N v . A graph G is said to be distance-balanced [26] if its each edge is distance-balanced. It is known that a connected graph G is transmission regular if and only if G is distance balanced [3], [26].
The fundamental properties of Wiener index and their extremal graphs are summarized in [9], [12], [16], [13], [21]. Transmission regular graphs are characterized by the following property: , [26], [29]). Let G be a connected graph with n vertices and m edges. Then with equality if and only if G is transmission regular.
Lemma 6 ( [3], [12]). Let G be a connected graph and let uv be an edge of G. Then Lemma 7. Let G be a connected graph. Then the following hold:

In (i), (ii) and (iii) the equality holds if and only if G is transmission regular.
Proof. (i) is a direct consequence of Lemma 6.
(iv) It follows from the proof of part (ii) and (iii) that uv∈E(G) Remark 5. Based on Lemma 7, the transmission-based topological index QS v,e (G) can be represented in the following alternative form: Proposition 5. Let G be a connected graph with n vertices and m edges. Then with equality if and only if G is a bipartite graph.
Proof. Let G be a connected graph with n vertices. It follows from Remark 4 (i) that for any edge uv of G, N u + N v ≤ n, with equality if and only if G is bipartite. This implies that with equality if and only if G is bipartite. Consequently, by Lemma 7 (iv) we have with equality if and only if G is bipartite.
Proposition 6. Let G be a connected graph with n vertices. Then with equality if and only if G is a bipartite graph.
Proof. Let G be a connected graph with n vertices. It follows from Remark 4 (i) that for any edge uv of G, N u + N v ≤ n, with equality if and only if G is bipartite. Therefore, it follows from Lemma 6 and with equality if and only if G is bipartite. This implies that Eq. (8) holds with equality if and only if G is bipartite.
Corollary 4. Let T n be an n vertex tree. Then Proof. It is a consequence of Proposition 5, Proposition 6 and Remark 4, since a tree with n vertices is bipartite and has exactly n − 1 edges.

Proposition 7 ( [12]
). Let G B be a connected bipartite graph with n vertices and m edges. Then with equality if and only if G is transmission regular.  Proof. Using Cauchy-Schwartz inequality and Proposition 7 one obtains for G B that with equality if and only if |σ(u) − σ(v)| is constant for any edge uv ∈ G B . Consequently, with equality if and only if |σ(u) − σ(v)| is constant for any edge uv ∈ G B . Because with equality if and only if |σ(u) − σ(v)| is constant for any edge uv ∈ G B .

The following proposition demonstrates that the Wiener index and the first transmission
Zagreb index are closely related.
Proposition 8. Let T n be an n-vertex tree. Then Proof. For any connected graph G we have Therefore, by Lemma 8 the result follows.
Remark 6. As a consequence of Eq. (9), we conclude that in the family of n-vertex trees there is a linear correspondence (a perfect linear correlation) between the topological indices W (T n ) and M S 1 (T n ).
In [40] it is reported that for a connected graph G, W (G) < M S 1 (G). This relation can be strengthened as follows: Proposition 9. Let G be a connected graph with minimum degree δ and maximum degree ∆. Then and equalities hold in both sides if and only if G is a regular graph.
Corollary 6. Let T n be an n-vertex tree. Then where (i) the right-hand side equality holds if and only if T n is the path P n ; (ii) the left-hand side equality holds if and only if T n is the star S n .
Proof. For an n-vertex tree T n we have W (S n ) ≤ W (T n ) ≤ W (P n ), where W (S n ) = (n − 1) 2 and W (P n ) = (n 3 − n) 6 . Therefore, from Proposition 8, we have the following inequalities: with equality if and only if T n is the path P n , and with equality if and only if T n is the star S n .
The following is a direct consequence of Proposition 9.
Corollary 7. If G be is a benzenoid graph with ∆ = 3 and δ = 2, then It is easy to show that the inequality represented by can be sharpened in the following form: Proposition 10. Let G be a connected graph with m edges. Then with equality if and only if |σ(u) − σ(v)| is constant for any uv ∈ E(G).
Proof. Using Cauchy-Schwartz inequality we have with equality if and only if |σ(u) − σ(v)| is constant for any uv ∈ E(G). It follows that with equality if and only if |σ(u) − σ(v)| is constant for any uv ∈ E(G). , [14]). Let G be a connected graph with n vertices and m edges. Then with equality if and only if diam(G) ≤ 2. (For example, the equality holds for complete graphs, complete bipartite and complete multipartite graphs, moreover wheel graphs and windmill graphs composed of triangles.) Proposition 11. Let G be a connected k-transmission regular with n vertices and m edges.
with equality if and only if diam(G) ≤ 2.
Proof. Since G is k-transmission regular, W (G) = 2k n . Now the result follows from Lemma 9.
Proposition 12. Let G be a connected graph with n vertices and m edges. Then and equalities hold in both sides simultaneously if diam(G) ≤ 2.
Proof. The result follows directly, using Lemma 9 and Proposition 2.
Proposition 13. Let G be a connected graph with n vertices and m edges. Then with equality if and only if σ(u) + σ(v) is constant for each edge uv ∈ E(G).
Proof. Using the Cauchy-Schwartz inequality, we obtain with equality if and only if σ(u) + σ(v) is constant for each edge uv ∈ E(G). This implies with equality if and only if σ(u) + σ(v) is constant for each edge uv ∈ E(G). Consequently, we have Let G be a connected graph with n vertices. Let us define the topological invariant Φ(G) as follows The following theorem shows that Φ(G) quantify the degree of transmission regularity of a connected graph G. Proof. Using Cauchy-Schwartz inequality, we obtain with equality if and only if σ(u) = σ(v) for each u, v ∈ V (G). This completes the proof.

Vertex and edge transitive graphs
In this section, following Darafshe [8], [34], we aim to use a method which applies group theory to graph theory. For more details regarding the theory of groups and graph theory one can see [15] and [19], respectively.
Let Γ be a group acting on a set X. We shall denote the action of α ∈ Γ on x ∈ X by x α . Then U ⊆ X is call an orbit of Γ on X if for every x, y ∈ U there exists α ∈ Γ such that x α = y. The action of group Γ on X is called transitive if X is itself an orbit of Γ on X.
Let G be a graph. A bijection α on V (G) is called an automorphism of G if it preserves E(G). In other words, α is an automorphism if for each u, v ∈ V (G), e = uv ∈ E(G) if and only if u α v α ∈ E(G). Let us denote by Aut(G) the set of all automorphisms of G. It is known that Aut(G) forms a group under the composition of mappings. This is a subgroup of the symmetric group on V (G). Note that Aut(G) acts on V (G) naturally, i.e., for each α ∈ Aut(G) and v ∈ V (G) the action of α on v, v α , is defined as α(v). The action of Aut(G) on V (G) induces an action on E(G). In fact, for α ∈ Aut(G) and e = uv ∈ E(G), the action of α on e = uv, e α , is defined as u α v α .
A graph G is called vertex-transitive (edge-transitive) if the action of Aut(G) on V (G) (E(G)) is transitive.
Let G be a graph, V 1 , V 2 , . . . , V t be the orbits of Aut(G) under its natural action on V (G). Then for each 1 ≤ i ≤ t and for u, v ∈ V i , σ(u) = σ(v). In particular, if G is vertex transitive (t = 1), then for each u, v ∈ V (G), σ(u) = σ(v). Therefore vertex-transitive graphs are transmission regular. It is known that any vertex-transitive graph is (vertex degree) regular [19] and transmission regular [8], but note vise versa.
Lemma 10. Let G be a connected k-transmission regular graph with n vertices and m edges. Then Lemma 11. Let G be a connected vertex-transitive graph with n vertices and m edges and the valency r. Then , Proof. If G is a connected vertex-transitive graph with n vertices and m edges, then G is of valency r (r-regular) and k-transmission regular, for some natural numbers r and k. It follows that 2m = nr and 2W (G) = nk.
Fullerenes C n can be drawn for n = 20 and for all even n ≥ 24. They have n carbon atoms, 3n 2 bonds, 12 pentagonal and n 2 − 10 hexagonal faces. The most important member of the family of fullerenes is C 60 [30]. The smallest fullerene is C 20 . It is a well-known fact that among all fullerene graphs only C 20 and C 60 (see Figure 3) are vertex-transitive [18]. Since   Figure 4 and its 2-dimensional molecular graph is in Figure 5. It is regular of valency 3 and has pq vertices and 3pq 2 edges. It follows that |A j | 2 d(x j ). If G is vertex transitive then r = 1 and |A 1 | = |V(G)|. Therefore, W(G) = |V(G)| 2 d(x), for each vertex x. Apply our method on an toroidal fullerene (or achiral polyhex nanotorus) R = R[p, q], Figs. 2.5 and 2.6. To compute the Wiener index of this nanotorus, we first prove its molecular graph is vertex transitive.

Lemma 2 The molecular graph of a polyhex nanotorus is vertex transitive.
Proof To prove this lemma, we first notice that p and q must be even. Consider the vertices u ij and u rs of the molecular graph of a polyhex nanotori T = T[p, q], Fig. 2.6. Suppose both of i and r are odd or even and σ is a horizontal symmetry plane which maps u it to u rt , 1 ≤ t ≤ p and π is a vertical symmetry which maps u tj to u ts , 1 ≤ t ≤ q. Then σ and π are automorphisms of T and we have π σ(u ij ) = π(u rj ) = u rs . Thus u ij and u rs are in the same orbit under the action of Aut(G) on V(G). On the other hand, the map θ defined by θ(u ij ) = θ(u (p+1−i)j ) is a graph automorphism of T and so if "i is odd and r is even" or "i is even and r is odd" then again u ij and u rs will be in the same orbit of Aut(G), proving the lemma.   SJ(H n ) = n 2 2 2(n−1) (n2 n−1 − 2n + 2) √ n2 n , GAS(H n ) = n, HS(H n ) = 2n 2 2 2(n−1) .