Sharp bounds on the zeroth-order general Randić index of trees in terms of domination number

<abstract><p>The zeroth-order general Randić index of graph $ G = (V_G, E_G) $, denoted by $ ^0R_{\alpha}(G) $, is the sum of items $ (d_{v})^{\alpha} $ over all vertices $ v\in V_G $, where $ \alpha $ is a pertinently chosen real number. In this paper, we obtain the sharp upper and lower bounds on $ ^0R_{\alpha} $ of trees with a given domination number $ \gamma $, for $ \alpha\in(-\infty, 0)\cup(1, \infty) $ and $ \alpha\in(0, 1) $, respectively. The corresponding extremal graphs of these bounds are also characterized.</p></abstract>

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Some Bounds on Zeroth-Order General Randić Index

Mathematics ◽

10.3390/math8010098 ◽

2020 ◽

Vol 8 (1) ◽

pp. 98 ◽

Cited By ~ 2

Author(s):

Muhammad Kamran Jamil ◽

Ioan Tomescu ◽

Muhammad Imran ◽

Aisha Javed

Keyword(s):

Clique Number ◽

Upper Bounds ◽

Randić Index ◽

Extremal Graphs ◽

Lower And Upper Bounds ◽

Zeroth Order ◽

Randic Index ◽

General Randić Index ◽

Inverse Degree ◽

General Randic Index

For a graph G without isolated vertices, the inverse degree of a graph G is defined as I D ( G ) = ∑ u ∈ V ( G ) d ( u ) − 1 where d ( u ) is the number of vertices adjacent to the vertex u in G. By replacing − 1 by any non-zero real number we obtain zeroth-order general Randić index, i.e., 0 R γ ( G ) = ∑ u ∈ V ( G ) d ( u ) γ , where γ ∈ R − { 0 } . Xu et al. investigated some lower and upper bounds on I D for a connected graph G in terms of connectivity, chromatic number, number of cut edges, and clique number. In this paper, we extend their results and investigate if the same results hold for γ < 0 . The corresponding extremal graphs have also been identified.

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Some Upper Bounds on the First General Zagreb Index

Journal of Mathematics ◽

10.1155/2022/8131346 ◽

2022 ◽

Vol 2022 ◽

pp. 1-4

Author(s):

Muhammad Kamran Jamil ◽

Aisha Javed ◽

Ebenezer Bonyah ◽

Iqra Zaman

Keyword(s):

Minimum Degree ◽

Upper Bounds ◽

Upper And Lower Bounds ◽

Randić Index ◽

Zeroth Order ◽

Zagreb Index ◽

Randic Index ◽

Independent Number ◽

General Randić Index ◽

General Randic Index

The first general Zagreb index M γ G or zeroth-order general Randić index of a graph G is defined as M γ G = ∑ v ∈ V d v γ where γ is any nonzero real number, d v is the degree of the vertex v and γ = 2 gives the classical first Zagreb index. The researchers investigated some sharp upper and lower bounds on zeroth-order general Randić index (for γ < 0 ) in terms of connectivity, minimum degree, and independent number. In this paper, we put sharp upper bounds on the first general Zagreb index in terms of independent number, minimum degree, and connectivity for γ . Furthermore, extremal graphs are also investigated which attained the upper bounds.

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On the zeroth-order general Randić index, variable sum exdeg index and trees having vertices with prescribed degree

Discrete Mathematics Algorithms and Applications ◽

10.1142/s1793830918500155 ◽

2018 ◽

Vol 10 (02) ◽

pp. 1850015 ◽

Cited By ~ 1

Author(s):

Sohaib Khalid ◽

Akbar Ali

Keyword(s):

Real Number ◽

Fixed Number ◽

Randić Index ◽

Zeroth Order ◽

Positive Real ◽

Randic Index ◽

General Randić Index ◽

General Randic Index ◽

Pendent Vertices ◽

Number Of Segments

The zeroth-order general Randić index (usually denoted by [Formula: see text]) and variable sum exdeg index (denoted by [Formula: see text]) of a graph [Formula: see text] are defined as [Formula: see text] and [Formula: see text], respectively, where [Formula: see text] is degree of the vertex [Formula: see text], [Formula: see text] is a positive real number different from 1 and [Formula: see text] is a real number other than [Formula: see text] and [Formula: see text]. A segment of a tree is a path [Formula: see text], whose terminal vertices are branching or/and pendent, and all non-terminal vertices (if exist) of [Formula: see text] have degree 2. For [Formula: see text], let [Formula: see text], [Formula: see text], [Formula: see text] be the collections of all [Formula: see text]-vertex trees having [Formula: see text] pendent vertices, [Formula: see text] segments, [Formula: see text] branching vertices, respectively. In this paper, all the trees with extremum (maximum and minimum) zeroth-order general Randić index and variable sum exdeg index are determined from the collections [Formula: see text], [Formula: see text], [Formula: see text]. The obtained extremal trees for the collection [Formula: see text] are also extremal trees for the collection of all [Formula: see text]-vertex trees having fixed number of vertices with degree 2 (because the number of segments of a tree [Formula: see text] can be determined from the number of vertices of [Formula: see text] having degree 2 and vice versa).

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