Comparing the Zagreb indices for graphs with small difference between the maximum and minimum degrees

Sun, Lingli; Chen, Ting

doi:10.1016/j.dam.2008.10.007

Cited by 11 publications

(17 citation statements)

References 9 publications

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“…We encourage the reader to consult papers [32][33][34][35] for the mathematical properties of eccentric connectivity index. Some recent results on the Zagreb indices are reported in [36][37][38][39][40][41][42][43][44]. In this report we compare the eccentric connectivity index (ξ C ) and the Zagreb indices (M 1 and M 2 ) for chemical trees.…”

Section: Introductionmentioning

confidence: 91%

Relationship between the eccentric connectivity index and Zagreb indices

Das

Trinajstić

2011

Computers & Mathematics with Applications

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Eccentric connectivity index (ξ C ) First Zagreb index (M 1 ) Second Zagreb index (M 2 ) a b s t r a c t For a (molecular) graph, the first Zagreb index M 1 is equal to the sum of the squares of the degrees of the vertices, and the second Zagreb index M 2 is equal to the sum of the products of the degrees of pairs of adjacent vertices. If G is a connected graph with vertex set V (G), then the eccentric connectivity index ofwhere d i is the degree of a vertex v i and e i is its eccentricity. In this report we compare the eccentric connectivity index (ξ C ) and the Zagreb indices (M 1 and M 2 ) for chemical trees. Moreover, we compare the eccentric connectivity index (ξ C ) and the first Zagreb index (M 1 ) for molecular graphs.

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Section: Introductionmentioning

confidence: 91%

Relationship between the eccentric connectivity index and Zagreb indices

Das

Trinajstić

2011

Computers & Mathematics with Applications

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“…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 generalized version of [18] was considered for various classes of graphs. These studies are summarized in two recent surveys [23,24].…”

Section: Comparing Zagreb Indicesmentioning

confidence: 99%

On the Zagreb indices equality

Abdo

Dimitrov

Gutman

2012

Discrete Applied Mathematics

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For a simple graph G with n vertices and m edges, the first Zagreb index and the second Zagreb index are defined asIn [34], 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 ∆ = 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.

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“…It is known that all intervals of lengths 1, 2, 3, and 4, except [2,5], are good, see [13]. In [1], it was shown that for every integer c, the interval [c, c + ⌈ √ c⌉] is good.…”

Section: Good Intervalsmentioning

confidence: 99%

“…Now, we prove the opposite direction. For n = 0, 1, 2 and an arbitrary integer a, all intervals [a, a + n] are good, see [13], and they satisfy the inequality a ≥ n(n−1) 2 . For n = 3, [1,4] is the only good interval that does not satisfy a ≥ n(n−1)…”

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confidence: 99%

On the Zagreb index inequality of graphs with prescribed vertex degrees

Andova

Bogoev

Dimitrov

et al. 2011

Discrete Applied Mathematics

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a b s t r a c tFor a simple graph G with n vertices and m edges, the inequality M 1 (G)/n ≤ M 2 (G)/m, where M 1 (G) and M 2 (G) are the first and the second Zagreb indices of G, is known as the Zagreb indices inequality. A set S is good if for every graph whose degrees of vertices are in S, the inequality holds. We characterize that an interval [a, a + n] is good if and only if a ≥ n(n−1) 2 or [a, a + n] = [1, 4]. We also present an algorithm that decides if an arbitrary set S of cardinality s is good, which requires O(s 2 log s) time and O(s) space.

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Comparing the Zagreb indices for graphs with small difference between the maximum and minimum degrees

Cited by 11 publications

References 9 publications

Relationship between the eccentric connectivity index and Zagreb indices

Relationship between the eccentric connectivity index and Zagreb indices

On the Zagreb indices equality

On the Zagreb index inequality of graphs with prescribed vertex degrees

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