scholarly journals Sharp Lower Bounds of the Sum-Connectivity Index of Unicyclic Graphs

2021 ◽  
Vol 2021 ◽  
pp. 1-6
Author(s):  
Maryam Atapour
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Connectivity Index ◽  
Unicyclic Graphs ◽  
Sharp Lower Bound

The sum-connectivity index of a graph G is defined as the sum of weights 1 / d u + d v over all edges u v of G , where d u and d v are the degrees of the vertices u and v in graph G , respectively. In this paper, we give a sharp lower bound on the sum-connectivity index unicyclic graphs of order n ≥ 7 and diameter D G ≥ 5 .

Sharp lower bounds on the sum-connectivity index of quasi-tree graphs

2021 ◽  
pp. 2150136
Author(s):  
Amir Taghi Karimi
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Connectivity Index ◽  
Tree Graphs ◽  
Sharp Lower Bound

The sum-connectivity index of a graph [Formula: see text] is defined as the sum of weights [Formula: see text] over all edges [Formula: see text] of [Formula: see text], where [Formula: see text] and [Formula: see text] are the degrees of the vertices [Formula: see text] and [Formula: see text] in [Formula: see text], respectively. A graph [Formula: see text] is called quasi-tree, if there exists [Formula: see text] such that [Formula: see text] is a tree. In the paper, we give a sharp lower bound on the sum-connectivity index of quasi-tree graphs.

Two-tree graphs with minimum sum-connectivity index

2020 ◽  
pp. 2150053
Author(s):  
A. Jahanbani
Keyword(s):  
Lower Bound ◽  
Complete Graph ◽  
Connectivity Index ◽  
Tree Graphs ◽  
Sharp Lower Bound

The sum-connectivity index of a graph [Formula: see text] is defined as the sum of weights [Formula: see text] over all edges [Formula: see text] of [Formula: see text], where [Formula: see text] and [Formula: see text] are the degrees of the vertices [Formula: see text] and [Formula: see text] in [Formula: see text], respectively. The graphs called two-trees are defined by recursion. The smallest two-tree is the complete graph on two vertices. A two-tree on [Formula: see text] vertices (where [Formula: see text]) is obtained by adding a new vertex adjacent to the two end vertices of one edge in a two-tree on [Formula: see text] vertices. In this paper, the sharp lower bound on the sum-connectivity index of two-trees is presented, and the two-trees with the minimum and the second minimum sum-connectivity, respectively, are determined.

General sum-connectivity index of unicyclic graphs with given diameter and girth

2021 ◽  
pp. 2150140
Author(s):  
Tomáš Vetrík
Keyword(s):  
Lower Bounds ◽  
Topological Indices ◽  
Connectivity Index ◽  
Index Formula ◽  
Extremal Graphs ◽  
Unicyclic Graphs ◽  
Harmonic Index

Topological indices of graphs have been studied due to their extensive applications in chemistry. We obtain lower bounds on the general sum-connectivity index [Formula: see text] for unicyclic graphs [Formula: see text] of given girth and diameter, and for unicyclic graphs of given diameter, where [Formula: see text]. We present the extremal graphs for all the bounds. Our results generalize previously known results on the harmonic index for unicyclic graphs of given diameter.

scholarly journals Monomial ideals and the failure of the Strong Lefschetz property

2021 ◽  
Author(s):  
Nasrin Altafi ◽  
Samuel Lundqvist
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Hilbert Function ◽  
Monomial Ideal ◽  
Monomial Ideals ◽  
Hilbert Functions ◽  
Lefschetz Property ◽  
Strong Lefschetz Property ◽  
Sharp Lower Bound

AbstractWe give a sharp lower bound for the Hilbert function in degree d of artinian quotients $$\Bbbk [x_1,\ldots ,x_n]/I$$ k [ x 1 , … , x n ] / I failing the Strong Lefschetz property, where I is a monomial ideal generated in degree $$d \ge 2$$ d ≥ 2 . We also provide sharp lower bounds for other classes of ideals, and connect our result to the classification of the Hilbert functions forcing the Strong Lefschetz property by Zanello and Zylinski.

scholarly journals On the eccentric connectivity coindex in graphs

2021 ◽  
Vol 7 (1) ◽  
pp. 651-666
Author(s):  
Hongzhuan Wang ◽  
◽  
Xianhao Shi ◽  
Ber-Lin Yu
Keyword(s):  
Lower Bounds ◽  
Connected Graph ◽  
Topological Index ◽  
Extremal Problems ◽  
Connectivity Index ◽  
Adjacent Vertex ◽  
Extremal Graph ◽  
Eccentric Connectivity Index ◽  
Unicyclic Graphs

<abstract><p>The well-studied eccentric connectivity index directly consider the contribution of all edges in a graph. By considering the total eccentricity sum of all non-adjacent vertex, Hua et al. proposed a new topological index, namely, eccentric connectivity coindex of a connected graph. The eccentric connectivity coindex of a connected graph $ G $ is defined as</p> <p><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \overline{\xi}^{c}(G) = \sum\limits_{uv\notin E(G)} (\varepsilon_{G}(u)+\varepsilon_{G}(v)). $\end{document} </tex-math></disp-formula></p> <p>Where $ \varepsilon_{G}(u) $ (resp. $ \varepsilon_{G}(v) $) is the eccentricity of the vertex $ u $ (resp. $ v $). In this paper, some extremal problems on the $ \overline{\xi}^{c} $ of graphs with given parameters are considered. We present the sharp lower bounds on $ \overline{\xi}^{c} $ for general connecteds graphs. We determine the smallest eccentric connectivity coindex of cacti of given order and cycles. Also, we characterize the graph with minimum and maximum eccentric connectivity coindex among all the trees with given order and diameter. Additionally, we determine the smallest eccentric connectivity coindex of unicyclic graphs with given order and diameter and the corresponding extremal graph is characterized as well.</p></abstract>

Lower Bounds for Matrices, II

1991 ◽  
Vol 44 (1) ◽  
pp. 54-74 ◽  
Author(s):  
Grahame Bennett
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Upper Bound ◽  
Monotonicity Property ◽  
Hausdorff Matrix ◽  
Increasing Function ◽  
Sharp Lower Bound

AbstractOur main result is the following monotonicity property for moment sequences μ. Let p be fixed, 1 ≤ p < ∞: thenis an increasing function of r(r = 1,2,…). From this we derive a sharp lower bound for an arbitrary Hausdorff matrix acting on ℓp.The corresponding upper bound problem was solved by Hardy.

The least distance between extremums and minimal period of solutions to autonomous vector differential equations

2019 ◽  
Vol 485 (2) ◽  
pp. 142-144
Author(s):  
A. A. Zevin
Keyword(s):  
Lower Bound ◽  
Differential Equations ◽  
Periodic Solutions ◽  
Lipschitz Constant ◽  
Minimal Period ◽  
Arbitrary Vector ◽  
Vector Norm ◽  
Sharp Lower Bound

Solutions x(t) of the Lipschitz equation x = f(x) with an arbitrary vector norm are considered. It is proved that the sharp lower bound for the distances between successive extremums of xk(t) equals π/L where L is the Lipschitz constant. For non-constant periodic solutions, the lower bound for the periods is 2π/L. These estimates are achieved for norms that are invariant with respect to permutation of the indices.

A Lower Bound for Schur Numbers and Multicolor Ramsey Numbers

10.37236/1188 ◽  
1994 ◽  
Vol 1 (1) ◽  
Author(s):  
Geoffrey Exoo
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Ramsey Numbers

For $k \geq 5$, we establish new lower bounds on the Schur numbers $S(k)$ and on the k-color Ramsey numbers of $K_3$.

scholarly journals A Lower Bound for the Query Phase of Contraction Hierarchies and Hub Labels and a Provably Optimal Instance-Based Schema

Algorithms ◽  
2021 ◽  
Vol 14 (6) ◽  
pp. 164
Author(s):  
Tobias Rupp ◽  
Stefan Funke
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Real World ◽  
Road Network ◽  
Upper And Lower Bounds ◽  
Bounded Degree ◽  
Shortest Path Routing ◽  
Graph Classes ◽  
Speed Up ◽  
To Come

We prove a Ω(n) lower bound on the query time for contraction hierarchies (CH) as well as hub labels, two popular speed-up techniques for shortest path routing. Our construction is based on a graph family not too far from subgraphs that occur in real-world road networks, in particular, it is planar and has a bounded degree. Additionally, we borrow ideas from our lower bound proof to come up with instance-based lower bounds for concrete road network instances of moderate size, reaching up to 96% of an upper bound given by a constructed CH. For a variant of our instance-based schema applied to some special graph classes, we can even show matching upper and lower bounds.

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