Sharp lower bounds on the sum-connectivity index of trees

2021 ◽  
Author(s):  
S. Alyar ◽  
R. Khoeilar
Keyword(s):  
Lower Bounds ◽  
Domination Number ◽  
Extremal Problems ◽  
Connectivity Index ◽  
Matching Number ◽  
Extremal Graphs ◽  
Extremal Values

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. In this paper, some extremal problems on the sum-connectivity index of trees are studied. The extremal values on the sum-connectivity index of trees with given graphic parameters, such as pendant number, matching number, domination number and diameter, are determined. The corresponding extremal graphs are characterized, respectively.

On the domination number of a graph and its total graph

2020 ◽  
Vol 12 (05) ◽  
pp. 2050068
Author(s):  
E. Murugan ◽  
J. Paulraj Joseph
Keyword(s):  
Lower Bounds ◽  
Domination Number ◽  
Upper And Lower Bounds ◽  
Extremal Graphs ◽  
Total Graph

In this paper, we investigate the upper and lower bounds for the sum of domination number of a graph and its total graph and characterize the extremal graphs.

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.

Bounds on 2-point set domination number of a graph

2018 ◽  
Vol 10 (01) ◽  
pp. 1850012
Author(s):  
Purnima Gupta ◽  
Deepti Jain
Keyword(s):  
Lower Bounds ◽  
Upper Bound ◽  
Nonempty Subset ◽  
Dominating Set ◽  
Domination Number ◽  
Extremal Graphs ◽  
Minimum Cardinality ◽  
Point Set

A set [Formula: see text] is a [Formula: see text]-point set dominating set (2-psd set) of a graph [Formula: see text] if for any subset [Formula: see text], there exists a nonempty subset [Formula: see text] containing at most two vertices such that the subgraph [Formula: see text] induced by [Formula: see text] is connected. The [Formula: see text]-point set domination number of [Formula: see text], denoted by [Formula: see text], is the minimum cardinality of a 2-psd set of [Formula: see text]. In this paper, we determine the lower bounds and an upper bound on [Formula: see text] of a graph. We also characterize extremal graphs for the lower bounds and identify some well-known classes of both separable and nonseparable graphs attaining the upper bound.

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

2021 ◽  
Vol 7 (2) ◽  
pp. 2529-2542
Author(s):  
Chang Liu ◽  
◽  
Jianping Li
Keyword(s):  
Real Number ◽  
Lower Bounds ◽  
Domination Number ◽  
Upper And Lower Bounds ◽  
Randić Index ◽  
Extremal Graphs ◽  
Zeroth Order ◽  
Randic Index ◽  
General Randić Index ◽  
General Randic Index

<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>

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>

scholarly journals (β ,α)−Connectivity Index of Graphs

2017 ◽  
Vol 2 (1) ◽  
pp. 21-30 ◽  
Author(s):  
B. Basavanagoud ◽  
Veena R. Desai ◽  
Shreekant Patil
Keyword(s):  
Real Number ◽  
Lower Bounds ◽  
Connectivity Index ◽  
Upper And Lower Bounds ◽  
Line Graphs ◽  
Extremal Graphs ◽  
Acentric Factor ◽  
Wheel Graphs ◽  
Octane Isomers ◽  
Ladder Graphs

AbstractLet Eβ (G) be the set of paths of length β in a graph G. For an integer β ≥ 1 and a real number α, the (β,α)-connectivity index is defined as$$\begin{array}{} \displaystyle ^\beta\chi_\alpha(G)=\sum \limits_{v_1v_2 \cdot \cdot \cdot v_{\beta+1}\in E_\beta(G)}(d_{G}(v_1)d_{G}(v_2)...d_{G}(v_{\beta+1}))^{\alpha}. \end{array}$$The (2,1)-connectivity index shows good correlation with acentric factor of an octane isomers. In this paper, we compute the (2, α)-connectivity index of certain class of graphs, present the upper and lower bounds for (2, α)-connectivity index in terms of number of vertices, number of edges and minimum vertex degree and determine the extremal graphs which achieve the bounds. Further, we compute the (2, α)-connectivity index of line graphs of subdivision graphs of 2D-lattice, nanotube and nanotorus of TUC4C8[p,q], tadpole graphs, wheel graphs and ladder graphs.

scholarly journals On the Generalized Distance Energy of Graphs

Mathematics ◽  
2019 ◽  
Vol 8 (1) ◽  
pp. 17 ◽  
Author(s):  
Abdollah Alhevaz ◽  
Maryam Baghipur ◽  
Hilal A. Ganie ◽  
Yilun Shang
Keyword(s):  
Lower Bounds ◽  
Complete Graph ◽  
Connected Graph ◽  
Distance Matrix ◽  
Wiener Index ◽  
Diagonal Matrix ◽  
Upper And Lower Bounds ◽  
Star Graph ◽  
Extremal Graphs ◽  
Generalized Distance

The generalized distance matrix D α ( G ) of a connected graph G is defined as D α ( G ) = α T r ( G ) + ( 1 − α ) D ( G ) , where 0 ≤ α ≤ 1 , D ( G ) is the distance matrix and T r ( G ) is the diagonal matrix of the node transmissions. In this paper, we extend the concept of energy to the generalized distance matrix and define the generalized distance energy E D α ( G ) . Some new upper and lower bounds for the generalized distance energy E D α ( G ) of G are established based on parameters including the Wiener index W ( G ) and the transmission degrees. Extremal graphs attaining these bounds are identified. It is found that the complete graph has the minimum generalized distance energy among all connected graphs, while the minimum is attained by the star graph among trees of order n.

scholarly journals Multiplicative Zagreb indices of cacti

2016 ◽  
Vol 08 (03) ◽  
pp. 1650040 ◽  
Author(s):  
Shaohui Wang ◽  
Bing Wei
Keyword(s):  
Graph Theory ◽  
Lower Bounds ◽  
Connected Graph ◽  
Upper And Lower Bounds ◽  
Extremal Graphs ◽  
Zagreb Index ◽  
Zagreb Indices ◽  
Material Chemistry ◽  
Cactus Graph ◽  
Cactus Graphs

Let [Formula: see text] be multiplicative Zagreb index of a graph [Formula: see text]. A connected graph is a cactus graph if and only if any two of its cycles have at most one vertex in common, which is a generalization of trees and has been the interest of researchers in the field of material chemistry and graph theory. In this paper, we use a new tool to obtain the upper and lower bounds of [Formula: see text] for all cactus graphs and characterize the corresponding extremal graphs.

scholarly journals On k-rainbow domination in middle graphs

2021 ◽  
Author(s):  
Kijung Kim
Keyword(s):  
Minimum Weight ◽  
Domination Number ◽  
Simple Graph ◽  
Upper And Lower Bounds ◽  
Matching Number ◽  
Domatic Number ◽  
Rainbow Domination ◽  
Vertex Set ◽  
Middle Graph ◽  
Dominating Function

Let $G$ be a finite simple graph with vertex set $V(G)$ and edge set $E(G)$. A function $f : V(G) \rightarrow \mathcal{P}(\{1, 2, \dotsc, k\})$ is a \textit{$k$-rainbow dominating function} on $G$ if for each vertex $v \in V(G)$ for which $f(v)= \emptyset$, it holds that $\bigcup_{u \in N(v)}f(u) = \{1, 2, \dotsc, k\}$. The weight of a $k$-rainbow dominating function is the value $\sum_{v \in V(G)}|f(v)|$. The \textit{$k$-rainbow domination number} $\gamma_{rk}(G)$ is the minimum weight of a $k$-rainbow dominating function on $G$. In this paper, we initiate the study of $k$-rainbow domination numbers in middle graphs. We define the concept of a middle $k$-rainbow dominating function, obtain some bounds related to it and determine the middle $3$-rainbow domination number of some classes of graphs. We also provide upper and lower bounds for the middle $3$-rainbow domination number of trees in terms of the matching number. In addition, we determine the $3$-rainbow domatic number for the middle graph of paths and cycles.

scholarly journals Distance Domination and Distance Irredundance in Graphs

10.37236/953 ◽  
2007 ◽  
Vol 14 (1) ◽  
Author(s):  
Adriana Hansberg ◽  
Dirk Meierling ◽  
Lutz Volkmann
Keyword(s):  
Lower Bounds ◽  
Connected Graph ◽  
Dominating Set ◽  
Domination Number ◽  
Dominating Sets ◽  
Minimum Cardinality ◽  
Distance Domination

A set $D\subseteq V$ of vertices is said to be a (connected) distance $k$-dominating set of $G$ if the distance between each vertex $u\in V-D$ and $D$ is at most $k$ (and $D$ induces a connected graph in $G$). The minimum cardinality of a (connected) distance $k$-dominating set in $G$ is the (connected) distance $k$-domination number of $G$, denoted by $\gamma_k(G)$ ($\gamma_k^c(G)$, respectively). The set $D$ is defined to be a total $k$-dominating set of $G$ if every vertex in $V$ is within distance $k$ from some vertex of $D$ other than itself. The minimum cardinality among all total $k$-dominating sets of $G$ is called the total $k$-domination number of $G$ and is denoted by $\gamma_k^t(G)$. For $x\in X\subseteq V$, if $N^k[x]-N^k[X-x]\neq\emptyset$, the vertex $x$ is said to be $k$-irredundant in $X$. A set $X$ containing only $k$-irredundant vertices is called $k$-irredundant. The $k$-irredundance number of $G$, denoted by $ir_k(G)$, is the minimum cardinality taken over all maximal $k$-irredundant sets of vertices of $G$. In this paper we establish lower bounds for the distance $k$-irredundance number of graphs and trees. More precisely, we prove that ${5k+1\over 2}ir_k(G)\geq \gamma_k^c(G)+2k$ for each connected graph $G$ and $(2k+1)ir_k(T)\geq\gamma_k^c(T)+2k\geq |V|+2k-kn_1(T)$ for each tree $T=(V,E)$ with $n_1(T)$ leaves. A class of examples shows that the latter bound is sharp. The second inequality generalizes a result of Meierling and Volkmann and Cyman, Lemańska and Raczek regarding $\gamma_k$ and the first generalizes a result of Favaron and Kratsch regarding $ir_1$. Furthermore, we shall show that $\gamma_k^c(G)\leq{3k+1\over2}\gamma_k^t(G)-2k$ for each connected graph $G$, thereby generalizing a result of Favaron and Kratsch regarding $k=1$.

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