On the Zagreb Indices Equality

📝 Abstract

For a simple graph $G$ with $n$ vertices and $m$ edges, the first Zagreb index and the second Zagreb index are defined as $M_1(G)=\sum_{v\in V}d(v)^2 $ and $M_2(G)=\sum_{uv\in E}d(u)d(v) $. In \cite{VGFAD}, it was shown that if a connected graph $G$ has maximal degree 4, then $G$ satisfies $M_1(G)/n = M_2(G)/m$ (also known as the Zagreb indices equality) if and only if $G$ is regular or biregular of class 1 (a biregular graph whose no two vertices of same degree are adjacent). There, it was also shown that there exist infinitely many connected graphs of maximal degree $\Delta= 5$ that are neither regular nor biregular of class 1 which satisfy the Zagreb indices equality. Here, we generalize that result by showing that there exist infinitely many connected graphs of maximal degree $\Delta \geq 5$ that are neither regular nor biregular graphs of class 1 which satisfy the Zagreb indices equality. We also consider when the above equality holds when the degrees of vertices of a given graph are in a prescribed interval of integers.

💡 Analysis

For a simple graph $G$ with $n$ vertices and $m$ edges, the first Zagreb index and the second Zagreb index are defined as $M_1(G)=\sum_{v\in V}d(v)^2 $ and $M_2(G)=\sum_{uv\in E}d(u)d(v) $. In \cite{VGFAD}, it was shown that if a connected graph $G$ has maximal degree 4, then $G$ satisfies $M_1(G)/n = M_2(G)/m$ (also known as the Zagreb indices equality) if and only if $G$ is regular or biregular of class 1 (a biregular graph whose no two vertices of same degree are adjacent). There, it was also shown that there exist infinitely many connected graphs of maximal degree $\Delta= 5$ that are neither regular nor biregular of class 1 which satisfy the Zagreb indices equality. Here, we generalize that result by showing that there exist infinitely many connected graphs of maximal degree $\Delta \geq 5$ that are neither regular nor biregular graphs of class 1 which satisfy the Zagreb indices equality. We also consider when the above equality holds when the degrees of vertices of a given graph are in a prescribed interval of integers.

📄 Content

On the Zagreb Indices Equality Hosam Abdoa, Darko Dimitrova, Ivan Gutmanb aInstitut f¨ur Informatik, Freie Universit¨at Berlin, Takustraße 9, D–14195 Berlin, Germany E-mail: [abdo,darko]@mi.fu-berlin.de bFaculty of Science, University of Kragujevac, P. O. Box 60, 34000 Kragujevac, Serbia E-mail: gutman@kg.ac.rs Abstract For a simple graph G with n vertices and m edges, the first Zagreb index and the second Zagreb index are defined as M1(G) = P v∈V d(v)2 and M2(G) = P uv∈E d(u)d(v). In [34], it was shown that if a connected graph G has maximal degree 4, then G satisfies M1(G)/n = M2(G)/m (also known as the Zagreb indices equality) if and only if G is regular or biregular of class 1 (a biregular graph whose no two vertices of same degree are adjacent). There, it was also shown that there exist infinitely many connected graphs of maximal degree ∆= 5 that are neither regular nor biregular of class 1 which satisfy the Zagreb indices equality. Here, we generalize that result by showing that there exist infinitely many connected graphs of maximal degree ∆≥5 that are neither regular nor biregular graphs of class 1 which satisfy the Zagreb indices equality. We also consider when the above equality holds when the degrees of vertices of a given graph are in a prescribed interval of integers. Keywords: first Zagreb index, second Zagreb index, comparing Zagreb indices 1 Introduction Let G = (V, E) be a simple graph with n = |V | vertices and m = |E| edges. For v ∈V , d(v) is its degree. The first Zagreb index M1(G) and the second Zagreb index M2(G) are defined as follows: M1(G) = X v∈V d(v)2 and M2(G) = X uv∈E d(u)d(v). For the sake of simplicity, we often use M1 and M2 instead of M1(G) and M2(G), respectively. In 1972 the quantities M1 and M2 were found to occur within certain approximate expressions for the total π-electron energy [16]. In 1975 these graph invariants were proposed to be measures of branching of the carbon-atom skeleton [15]. The name “Zagreb index” (or, more precisely, “Zagreb group index”) seems to be first used in the review article [4]. For details of the mathematical theory and chemical applications of the Zagreb indices see surveys [10, 14, 25, 30] and papers [12, 13, 36, 37, 38]. 1 arXiv:1106.1809v1 [cs.DM] 9 Jun 2011 2 We denote by Ka,b the complete bipartite graph with a vertices in one class and b vertices in the other one. Let D(G) be the set of the vertex degrees of G, i.e., D(G) = {d(v) | v ∈V }. The subdivision graph S(G) of a graph G is obtained by inserting a new vertex (of degree 2) on every edge of G. A regular graph is a graph where each vertex has the same degree. A regular graph with vertices of degree k is called a k-regular graph. The graph G is biregular if its vertex degrees assume exactly two distinct values. We distinguish between two types of biregular graphs: biregular graphs of class 1 have the property that no two vertices of the same degree are adjacent. In biregular graphs of class 2 at least one edge connects vertices of equal degree. Let G be a graph with n vertices and let a, b, and c be three positive integers, 1 ≤a ≤b ≤c ≤n−1. The graph G is said to be triregular if for i = 1, 2, . . . , n, either di = a or di = b or di = c, and there exists at least one vertex of degree a, at least one vertex of degree b, and at least one vertex of degree c. If so, then G is a triregular graph of degrees a, b, and c, or for brevity, an (a, b, c)-triregular graph. Similarly, as in the case of biregular graphs, we distinguish two types of triregular graphs: Triregular graphs of class 1 have the property that no two vertices of the same degree are adjacent. In triregular graphs of class 2 at least one edge connects vertices of equal degree. As defined in [1], a set S of integers is good if for every graph G with D(G) ⊆S, the inequality (1) holds. Otherwise, S is a bad set. 1.1 Comparing Zagreb indices In spite of the fact that the two Zagreb indices were introduced simultaneously and examined almost always together, relations between them were not considered until quite recently. Observe that, for general graphs, the order of magnitude of M1 is O(n3) while the order of magnitude of M2 is O(mn2). This suggests comparing M1/n and M2/m instead of M1 and M2. Based on his AutoGraphiX [6] conjecture-generating computer system, Pierre Hansen arrived at the inequality M1(G) n ≤M2(G) m (1) which he conjectured to hold for all connected graphs. In the current mathematico-chemical literature, the relation (1) is usually referred to as the Zagreb indices inequality. If the equality case is excluded, then we speak of the strict Zagreb indices inequality. Soon after the announcement of this conjecture it was shown [18] that there exist graphs for which (1) does not hold. Although the work [18] appeared to completely settle Hansen’s conjecture, it was just the beginning of a long series of studies [1, 2, 5, 8, 19, 21, 23, 26, 27, 32, 33] in which the validity or non-validity of either [18] or some generali

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