Unified extremal results of topological indices and spectral invariants of graphs

Yao, Yuedan; Liu, Muhuo; Belardo, Francesco; Yang, Chao

doi:10.1016/j.dam.2019.06.005

Cited by 20 publications

(7 citation statements)

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“…where f is a real-valued function. Some initial studies, recent developments, and a survey on the indices of the form (8) can be found in [23][24][25][26][27][28][29], respectively.…”

Section: Extremal Results Concerning Molecular ξ-Cyclic Graphsmentioning

confidence: 99%

Ad-Hoc Lanzhou Index

Ali,

Shang,

Dimitrov

et al. 2023

Mathematics

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This paper initiates the study of the mathematical aspects of the ad-hoc Lanzhou index. If G is a graph with the vertex set {x1,…,xn}, then the ad-hoc Lanzhou index of G is defined by Lz˜(G)=∑i=1ndi(n−1−di)2, where di represents the degree of the vertex xi. Several identities for the ad-hoc Lanzhou index, involving some existing topological indices, are established. The problems of finding graphs with the extremum values of the ad-hoc Lanzhou index from the following sets of graphs are also attacked: (i) the set of all connected ξ-cyclic graphs of a fixed order, (ii) the set of all connected molecular ξ-cyclic graphs of a fixed order, (iii) the set of all graphs of a fixed order, and (iv) the set of all connected molecular graphs of a fixed order.

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“…where f is a real-valued function. Some initial studies, recent developments, and a survey on the indices of the form (8) can be found in [23][24][25][26][27][28][29], respectively.…”

Section: Extremal Results Concerning Molecular ξ-Cyclic Graphsmentioning

confidence: 99%

Ad-Hoc Lanzhou Index

Ali,

Shang,

Dimitrov

et al. 2023

Mathematics

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“…When t ≥ 4, h(t) = 2h 1 (t) + h 2 (t)h 3 (t) − 2 ≥ 2h 1 (∞) + h 2 (4)h 3 (4) − 2 > 0. Thus, ∂g2 ∂r > h(t) ≥ 0 and g 2 (t, r) ≥ g 2 (t, 3) = h 6 (t) + 4h 7 (t) − 5, where h 6 (t) = dt ≥ 0 and h 9 (t) ≥ h 9 (1) = 0 on the interval [1,21]. When t ≥ 21, h 9 (t) ≥ h 6 (21) + 4h 7 (∞) − 5 > 0.…”

Section: In This Case Kmentioning

confidence: 99%

“…In [21], Yao, Liu, Belardo and Yang introduced the vertex-degree function-index H f (G) of a graph G with a real-valued function f (x) as follows:…”

Section: And Shimentioning

confidence: 99%

(n,m)-Graphs with Maximum Vertex–Degree Function–Index for Convex Functions

Xu,

2023

MATCH

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An (n, m)-graph is a graph with n vertices and m edges. The vertex-degree function-index H f (G) of a graph G is defined as, where f is a real function.In this paper, we show that if f (x) is strictly convex and strictly monotonically decreasing and satisfies some additional properties, thenThe unique graph that satisfies the above equality is characterized. As an instance, the function f (x) = (x + q) α is such a function when α ≤ −t, −1 < q ≤ 2.038t − 0.038 and t ≥ 1 or when α < 0, −1 < q ≤ 0.We also prove that if f (x) is strictly convex and strictly monotonically decreasing and satisfies some additional properties, thenThe unique graph that satisfies the above equality is characterized. As an instance, the function f (x) = (x + q) α has the properties as described above when α ≤ −t and 0 < q ≤ 1.413t + 0.587 and t ≥ 1.

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“…Further, Zheng et al [28] consider the extremal spectral radius of trees and unicyclic graphs by a unified method. Yao et al [23] found a unified method to identify the extremal graphs for a set of topological or spectral invariants by using the majorization theorem. It is worth mentioning that there is a review article [14] written by Li et al…”

Section: Introduction 1backgroundmentioning

confidence: 99%