On a linear combination of Zagreb indices

The modiﬁed ﬁrst Zagreb connection index of a triangle-free and quadrangle-free graph G is equal to 2 M 2 ( G ) − M 1 ( G ) , where M 2 ( G ) and M 1 ( G ) are the well-known second and ﬁrst Zagreb indices of G , respectively. This paper involves the study of the linear combination 2 M 2 ( G ) − M 1 ( G ) of M 2 ( G ) and M 1 ( G ) when G is a connected graph of a given order and cyclomatic number. More precisely, graphs having the minimum value of the graph invariant 2 M 2 − M 1 are determined from the class of all connected graphs of order n and cyclomatic number c y , when c y ≥ 1 and n ≥ 2( c y − 1) .


Introduction
Throughout this study, only finite and connected graphs are exclusively considered. For the undefined notations and concepts from graph theory, the readers are suggested to consult books [4,7,12].
A topological index associated with a graph is a number that does not change under graph isomorphism. In [2], a topological index appeared in [11] was thoroughly studied first time after its appearance. This index is the modified first Zagreb connection index [2], which is defined for a graph G as where V (G) represents the set of vertices of G, τ v is the number of those vertices of G that are at distance 2 from v, and d v denotes the degree of v. Zagreb connection indices have gained a considerable attention from mathematical community; for example see [6,8,[15][16][17].
The sum of squares of vertex degrees of a graph is often denoted by M 1 and is recognized as the first Zagreb index; for example, see [3,13]. The sum of the products of degrees of adjacent vertices of a graph is usually denoted by M 2 and is generally known as the second Zagreb index (see for example [18]). Thus, for a graph G, one has and where E(G) represents the set of edges of G. Additional information on Zagreb indices may be found in the review papers [5,10,14] and references cited therein. If a graph G is quadrangle-free and triangle-free, then Motivated from Equation (1), the linear combination 2M 2 − M 1 of M 2 and M 1 is studied in this paper for connected graphs of fixed order and cyclomatic number, where the cyclomatic number of a graph is the least number of edges whose removal * E-mail address: a.albalahi@uoh.edu.sa gives cycle-free graph. Since Equation (1) does not hold for infinitely many graphs containing triangle(s) or/and quadrangle(s), so when removing the conditions of being triangle-free and quadrangle-free from G, the right-hand side of Equation (1) is denoted by a modified notation: ZC † 1 . Thus, for any graph G, one has In the present study, graphs having the minimum value of the graph invariant ZC † 1 are determined from the class of all connected graphs of order n and cyclomatic number c y , when c y ≥ 1 and n ≥ 2(c y − 1). The case c y = 0 corresponds to trees, which has already been resolved in [2,9].

Main results
Before stating and proving the main results of this paper, some notations and definitions are recalled in the following. For then w is called a neighbor of v. A vertex in a graph is said to be a pendent vertex (branching vertex, respectively) if it has degree one (at least three, respectively). A non-trivial path P : u 1 u 2 · · · u t of a graph is called pendent path whenever min{d v1 , d vt } = 1 and max{d v1 , d vt } ≥ 3, and every remaining vertex of P (if it exists) has degree two. A graph with n vertices is also known as an n-vertex graph. Denote by G a graph deduced from some other graph G after utilizing a transformation provided that V (G ) = V (G). In the remaining part of this paper, wherever such types of graphs are considered simultaneously, the notion d v represents the degree of v ∈ V (G ) = V (G) in G.

Lemma 2.1. Every graph having the least value of ZC †
1 over the class of all n-vertex graphs with c y ≥ 1 cyclomatic number has the minimum degree at least 2.
Proof. Let G be a graph having the least value of ZC † 1 over the class of all n-vertex graphs with c y ≥ 1 cyclomatic number. Contrarily, suppose that the minimum degree of G is 1.
Since c y ≥ 1, at least one of the neighbors of v is non-pendent. Let u ∈ {v 1 } be a non-pendent neighbor of v. Denote by G the graph formed by dropping the edge uv from G and inserting there the edge v r u. It is obvious that the cyclomatic numbers of G and G are the same. When r = 1, one has which contradicts the definition of G. When r ≥ 2, one has  Proof. Take ∆ = ∆(G). Since only connected graphs are being considered and c y ≥ 2, one has ∆ ≥ 3. Suppose to the contrary that ∆ > 3. From Lemma 2.1, it follows that δ(G) ≥ 2. Since c y = |E(G)| − n + 1 and which implies that Equation (2) guaranties that G posses some vertex having degree 2. In the following, the vertex v ∈ V (G) is assumed to be a vertex of maximum degree, that is, d v = ∆.

Case 1.
At least one of the neighbors of v, say w, has degree 2.
Since ∆ > 3, there exists u ∈ N G (v) such that uw ∈ E(G). Let z ∈ {v} be the other neighbor of w. Denote by G the graph deduced from G after dropping vu and adding uw. Since δ ≥ 2 and d v = ∆ ≥ 4, one has a contradiction to the definition of G.

Case 2.
None of the neighbors of v has degree 2.
Since δ ≥ 2, every neighbor of v has degree at least 3. Recall that Equation (2) implies that G posses some vertex having degree 2. Since ∆ ≥ 4, there are vertices w ∈ V (G) \ N G (v) and v ∈ N G (v) such that d w = 2 and w v ∈ E(G). Denote by G the graph deduced from G after dropping vv and adding v w . Because every neighbor of v has degree at least 3 and For a graph G, define

Lemma 2.3. [1]
Let G be an n-vertex graph with cyclomatic number c y ≥ 2, minimum degree 2, and maximum degree 3.
(ii). If the inequality n > 5(c y − 1) holds then m 2,2 ≥ 1. Proof. By Lemmas 2.1, 2.2, and 2.3, one has δ(G) = 2, ∆(G) = 3, and m 2,2 ≥ 1. Take v, w ∈ V (G) such that d v = d w = 2 and vw ∈ E(G). Contrarily, assume that s, t ∈ V (G) are adjacent vertices of degree 3. Let x ∈ {v} be the neighbor of w. It is possible that x ∈ {s, t}; if it happens, then one takes x = t, without loss of generality. Case 1. The sets N G (v) and N G (w) are disjoint. Denote by G the graph deduced from G after dropping vw, wx, st and inserting vx, sw, tw. In either of the cases x = t and x = t, one has ZC † 1 (G) − ZC † 1 (G ) = 2, which contradicts the definition of G. Case 2. The sets N G (v) and N G (w) are not disjoint. Since G is connected and c y ≥ 2, it holds that d x = 3.
When x = t, then for the graph G obtained by dropping "wx, st" from G and inserting "wt, sx" there, one has Note that the sets N G (v) and N G (w) are disjoint and thus by Case 1, one gets a contradiction. When x = t, then let s 1 ∈ t be a neighbor of s. Denote by G the graph formed by dropping the edges wt, s 1 s and inserting the edges sw, s 1 t. Then, ZC † 1 (G) = ZC † 1 (G ). Again, the sets N G (v) and N G (w) are disjoint, and thus by Case 1, one has a contradiction. Proof. By Lemmas 2.1 and 2.2, one has δ(G) = 2 and ∆(G) = 3. It is claimed that max{m 2,2 , m 3,3 } = 0. Suppose to the contrary that max{m 2,2 , m 3,3 } > 0.
Lemma 2.6. For 2(c y − 1) ≤ n < 5(c y − 1) with c y ≥ 2, if G is a graph achieving the least value of ZC † 1 among all n-vertex graphs with c y cyclomatic number, then m 2,2 = 0.
Proof. Suppose to the contrary that m 2,2 > 0. By Lemmas 2.1, 2.2, and 2.3, one has δ(G) = 2, ∆(G) = 3, and m 3,3 ≥ 1. By the proof of Lemma 2.4 one deduces that there is an n-vertex graph G * with cyclomatic number c y such that ZC † 1 (G) > ZC † 1 (G * ), which contradicts the definition of G. Theorem 2.1. For c y ≥ 2, in the class of all n-vertex graphs with c y cyclomatic number, (i) the cubic graphs are the only graphs having the minimum ZC † 1 value whenever n = 2(c y − 1); (ii) the graphs of maximum degree 3 and minimum degree 2 with the constraint m 2,2 = 0, are the only graphs having the minimum ZC † 1 value whenever 2(c y − 1) < n < 5(c y − 1); (iii) the graphs of maximum degree 3 and minimum degree 2 with the constraint m 2,2 = m 3,3 = 0, are the only graphs having the minimum ZC † 1 value whenever n = 5(c y − 1); (iv) the graphs of maximum degree 3 and minimum degree 2 with the constraint m 3,3 = 0, are the only graphs having the minimum ZC † 1 value whenever n > 5(c y − 1).
Proof. Let G be a graph having the least value of ZC † 1 among all n-vertex graphs with c y cyclomatic number. By Lemmas 2.1 and 2.2, one has δ(G) = 2 and ∆(G) = 3. Thus, the following system of equations holds n 2 + n 3 = n, 2n 2 + 3n 3 = 2(n + c y − 1).