Abstract
The multiplicative first Zagreb index of a graph is defined as the product of the squares of the degrees of vertices of . The line graph of a graph is denoted by and is defined as the graph whose vertex set is the edge set of where two vertices of are adjacent if and only if they are adjacent in . The multiplicative first Zagreb index of the line graph of a graph is referred to as the reformulated multiplicative first Zagreb index of . This paper gives characterization of the unique graph attaining the minimum or maximum value of the reformulated multiplicative first Zagreb index in the class of all (i) trees of a fixed order (ii) connected unicyclic graphs of a fixed order.
1. Introduction
A numerical quantity of a graph that remains the same under graph isomorphism is known as a graph invariant. For a graph , denote by and , its vertex set and edge set, respectively. For a vertex , denote by and , the degree of and the set of all those vertices of that are adjacent to , respectively. The members of are known as neighbors of . For , the edge with its end-vertices and is denoted by and the degree of is defined as . In the following, we drop the symbol “” from “” whenever there is no confusion about the graph under consideration. Also, in the present paper, we consider only connected graphs. The notation and terminology used in this paper, but not defined here, can be found in standard (chemical) graph-theoretical books, such as [1–3].
Denote by the sum of the squares of the degrees of vertices of , that is,
The graph invariant , known as the first Zagreb index, was appeared in the first quarter of 1970s within a study of chemical modeling [4]. The first Zagreb index can be considered as one of the most studied graph invariants, especially in chemical graph theory [5–11].
The line graph of a graph is denoted by and is defined as the graph whose vertex set is the edge set of where two vertices of are adjacent if and only if they are adjacent in . The first Zagreb index of the line graph of a graph is known as the reformulated first Zagreb index [12] of , and for simplicity, it is denoted as . Thus,
Details about the mathematical properties of the reformulated first Zagreb index can be found in [13–16].
The multiplicative version of the first Zagreb index is defined [17, 18] as
The multiplicative first Zagreb index also gained a considerable attention from researchers, and hence, many investigations on this graph invariant have been conducted; for example, see [19, 20] and the related references included therein. Here, it should be pointed out that the multiplicative first Zagreb index is actually the square of the well-known Narumi–Katayama index [21], see, for example, [22], for details about the Narumi–Katayama index.
This paper is concerned with the graph invariant that is the product of the squares of the degrees of edges of . We call the invariant as the reformulated multiplicative first Zagreb index of and denote it by for simplicity. Thus,
Throughout this paper, whenever we consider a class of graphs of the same order, we assume that all the graphs of the considered class are pairwise nonisomorphic. A graph containing exactly one cycle is known as a unicyclic graph. By an -vertex graph, we mean a graph of order . For , denote by the class of all -vertex trees and let be the class of all -vertex connected unicyclic graphs. In this paper, the unique graph having the maximum or minimum value of the reformulated multiplicative first Zagreb index is characterized from each of the classes and for every fixed integer greater than 3.
2. Reformulated Multiplicative First Zagreb Index of Trees
This section is concerned with the characterization of the unique graph attaining the maximum or minimum value of the reformulated multiplicative first Zagreb index over the class of all trees of a fixed order .
A vertex of degree one is called a pendent vertex. A nontrivial path in a graph is said to be a pendent path if and , and every other vertex (if exists) of has degree 2. Two pendent paths having the same vertex of degree greater than 2 are known as adjacent pendent paths. By an ()‐graph, we mean a graph of order and size .
Lemma 1. If is a connected ()‐graph containing at least one pair of adjacent pendent paths, then there exists at least one connected ()-graph such that
Proof. Let and be two adjacent pendent paths in having lengths and , respectively. Without loss of generality, we assume that . Let be the graph deduced from by deleting the edge and inserting the edge (see Figure 1). Throughout this proof, by , we mean the degree of the vertex in . By the definition of the reformulated multiplicative first Zagreb index, there exists a positive integer such thatwhereFrom the inequality (which gives , that is, ) and equation (6), it follows that
Theorem 1. Among all trees of a fixed order , path is the unique graph with the minimum value of . In other words, if is any tree of order , thenwith equality if and only if is isomorphic to the path graph .
Proof. Let be a tree with the minimum value of among all trees of a fixed order . We claim that is isomorphic to . Contrarily, we assume that is not isomorphic to . Then, contains at least two adjacent pendent paths, and hence by Lemma 1, there exists at least one tree such thatwhich contradicts the definition of . Thus, .
Lemma 2. Let be a connected ()‐graph. Let be an edge such that it does not lie on any triangle of and , and . If is the graph formed from by deleting the edges and inserting the edges , then
Proof. In this proof, by , we mean the degree of the vertex in . By the definition of the reformulated multiplicative first Zagreb index, there exists a positive integer such thatSince , the right hand side of equation (12) is negative.
Theorem 2. Among all trees of a fixed order , star graph uniquely attains the maximum value of . In other words, if is any tree of order , thenwith equality if and only if is isomorphic to the star graph .
Proof. Let be a tree with the maximum value of among all trees of a fixed order . Let be the maximum degree of . We claim that is isomorphic to . Contrarily, we assume that is not isomorphic to . Then, , and hence by Lemma 2, there exists at least one -vertex tree such thatwhich contradicts the definition of . Thus, .
3. Reformulated Multiplicative First Zagreb Index of Unicyclic Graphs
This section is concerned with the maximum and minimum values of the reformulated multiplicative first Zagreb index over the class of all unicyclic graphs of a fixed order.
Theorem 3. Among all connected unicyclic graphs of a fixed order , the cycle graph attains uniquely the minimum value of . In other words, if is any connected unicyclic graphs of order , thenwith equality if and only if is isomorphic to .
Proof. Let be a graph attaining the minimum value of in the class of all connected unicyclic graphs of a fixed order . If contains at least two adjacent pendent paths, then by Lemma 1, there exists at least one connected unicyclic graph of order such thatwhich is a contradiction to the definition of . Thus, does not have any pair of adjacent pendent paths, which implies that the maximum degree of is at most 3.
We claim that the maximum degree of is 2, that is, . Contrarily, assume that is a pendent path in , where has degree 3 and has degree 1. Let be a neighbor of different from . Let be the graph formed from by deleting the edge and inserting the edge (see Figure 2). In the following, by , we mean the degree of the vertex in . By the definition of the reformulated multiplicative first Zagreb index, there exists a positive integer such thatwhereSince , it holds that ; this together with equation (17) implies thatwhich is again a contradiction to the definition of . Therefore, .
Theorem 4. Among all connected unicyclic graphs of a fixed order , the graph obtained from the star by adding an edge attains uniquely the maximum value of . In other words, if is any connected unicyclic graphs of order , thenwith equality if and only if is isomorphic to , where is the graph obtained from the star by adding an edge.
Proof. Let be a graph attaining the maximum value of in the class of all connected unicyclic graphs of a fixed order . From Lemma 2, it follows that the removal of all the vertices of degree 1 of results in (the cycle of order 3), that is, is isomorphic to the graph depicted in Figure 3. By routine calculations, one getswhere . Without loss of generality, we assume that . Thus, , and hence, . Equation (21) is equivalent towith the condition . For a fixed integer , since the function defined bywith and is strictly decreasing in and , we conclude that the maximum value of is attained if and only if . Therefore, is isomorphic to the graph .
Data Availability
The data used to support the findings of this study are available from the authors upon request.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This research has been funded by Scientific Research Deanship at University of Ha’il, Saudi Arabia, through project number RG-20050.