scholarly journals Two upper bounds for the degree distances of four sums of graphs

Filomat ◽  
2014 ◽  
Vol 28 (3) ◽  
pp. 579-590 ◽  
Author(s):  
Mingqiang An ◽  
Liming Xiong ◽  
Kinkar Das
Keyword(s):  
Connected Graph ◽  
Wiener Index ◽  
Upper Bounds ◽  
Vertex Degree ◽  
Weighted Sum ◽  
Degree Of A Vertex ◽  
Degree Distance

The degree distance (DD), which is a weight version of the Wiener index, defined for a connected graph G as vertex-degree-weighted sum of the distances, that is, DD(G) = ?{u,v}?V(G)[dG(u)+dG(v)]d[u,v|G), where dG(u) denotes the degree of a vertex u in G and d(u,v|G) denotes the distance between two vertices u and v in G: In this paper, we establish two upper bounds for the degree distances of four sums of two graphs in terms of other indices of two individual graphs.

scholarly journals On the reformulated reciprocal degree distance of graphs

2016 ◽  
Vol 25 (2) ◽  
pp. 205-213
Author(s):  
K. PATTABIRAMAN ◽  
◽  
M. VIJAYARAGAVAN ◽  
Keyword(s):  
Connected Graph ◽  
Cartesian Product ◽  
Vertex Degree ◽  
Symmetric Difference ◽  
Weighted Sum ◽  
Graph Invariant ◽  
Harary Index ◽  
Degree Distance

The reciprocal degree distance (RDD), defined for a connected graph G as vertex-degree-weighted sum of the reciprocal distances, that is, RDD(G) = P u,v∈V (G) (d(u)+d(v)) dG(u,v) . The new graph invariant named reformulated reciprocal degree distance is defined for a connected graph G as Rt(G) = P u,v∈V (G) (d(u)+d(v)) dG(u,v)+t , t ≥ 0. The reformulated reciprocal degree distance is a weight version of the t-Harary index, that is, Ht(G) = P u,v∈V (G) 1 dG(u,v)+t , t ≥ 0. In this paper, the reformulated reciprocal degree distance and reciprocal degree distance of disjunction, symmetric difference, Cartesian product of two graphs are obtained. Finally, we obtain the reformulated reciprocal degree distance and reciprocal degree distance of double a graph.

Upper bounds on product degree distance of F-sum of graphs

2019 ◽  
Vol 11 (04) ◽  
pp. 1950045
Author(s):  
K. Pattabiraman ◽  
Manzoor Ahmad Bhat
Keyword(s):  
Connected Graph ◽  
Upper Bounds ◽  
Degree Of A Vertex ◽  
Degree Distance ◽  
Appl Math

The product degree distance of a connected graph [Formula: see text] is defined as [Formula: see text] where [Formula: see text] is the degree of a vertex [Formula: see text] and [Formula: see text] is the distance between the vertices [Formula: see text] and [Formula: see text] in [Formula: see text] In this paper, we obtain two upper bounds for product degree distance of [Formula: see text]-sums of graphs which is defined by Eliasi [Four new sums of graphs and their Wiener indices, Discr. Appl. Math. 157 (2009) 794–803.]

scholarly journals Upper bounds on the energy of graphs in terms of matching number

2021 ◽  
pp. 16-16
Author(s):  
Saieed Akbari ◽  
Abdullah Alazemi ◽  
Milica Andjelic
Keyword(s):  
Adjacency Matrix ◽  
Connected Graph ◽  
Short Proof ◽  
Upper Bounds ◽  
Vertex Degree ◽  
Maximum Matching ◽  
Matching Number ◽  
Open Problems

The energy of a graph G, ?(G), is the sum of absolute values of the eigenvalues of its adjacency matrix. The matching number ?(G) is the number of edges in a maximum matching. In this paper, for a connected graph G of order n with largest vertex degree ? ? 6 we present two new upper bounds for the energy of a graph: ?(G) ? (n-1)?? and ?(G) ? 2?(G)??. The latter one improves recently obtained bound ?(G) ? {2?(G)?2?e + 1, if ?e is even; ?(G)(? a + 2?a + ?a-2?a), otherwise, where ?e stands for the largest edge degree and a = 2(?e + 1). We also present a short proof of this result and several open problems.

scholarly journals Upper Bounds for the Modified Second Multiplicative Zagreb Index of Graph Operations

2016 ◽  
Vol 17 ◽  
pp. 10-16
Author(s):  
Bommanahal Basavanagoud ◽  
Shreekant Patil
Keyword(s):  
Connected Graph ◽  
Cartesian Product ◽  
Upper Bounds ◽  
Zagreb Index ◽  
Graph Operations ◽  
Product Composition ◽  
Degree Of A Vertex ◽  
Product Of Graphs ◽  
Corona Product

The modified second multiplicative Zagreb index of a connected graph G, denoted by $\prod_{2}^{*}(G)$, is defined as $\prod_{2}^{*}(G)=\prod \limits_{uv\in E(G)}[d_{G}(u)+d_{G}(v)]^{[d_{G}(u)+d_{G}(v)]}$ where $d_{G}(z)$ is the degree of a vertex z in G. In this paper, we present some upper bounds for the modified second multiplicative Zagreb index of graph operations such as union, join, Cartesian product, composition and corona product of graphs are derived.The modified second multiplicative Zagreb index of aconnected graph , denoted by , is defined as where is the degree of avertex in . In this paper, we present some upper bounds for themodified second multiplicative Zagreb index of graph operations such as union,join, Cartesian product, composition and corona product of graphs are derived.

scholarly journals Some distance based indices of graphs based on four new operations related to the lexicographic product

2019 ◽  
Vol 11 (2) ◽  
pp. 258-267
Author(s):  
N. Dehgardi ◽  
S.M. Sheikholeslami ◽  
M. Soroudi
Keyword(s):  
Molecular Graph ◽  
Wiener Index ◽  
Lexicographic Product ◽  
Total Graph ◽  
Degree Of A Vertex ◽  
Degree Distance

For a (molecular) graph, the Wiener index, hyper-Wiener index and degree distance index are defined as $$W(G)= \sum_{\{u,v\}\subseteq V(G)}d_G(u,v),$$ $$WW(G)=W(G)+\sum_{\{u,v\}\subseteq V(G)} d_{G}(u,v)^2,$$ and $$DD(G)=\sum_{\{u,v\}\subseteq V(G)}d_G(u, v)(d(u/G)+d(v/G)),$$ respectively, where $d(u/G)$ denotes the degree of a vertex $u$ in $G$ and $d_G(u, v)$ is distance between two vertices $u$ and $v$ of a graph $G$. In this paper, we study Wiener index, hyper-Wiener index and degree distance index of graphs based on four new operations related to the lexicographic product, subdivision and total graph.

scholarly journals On the Sum of Distance Laplacian Eigenvalues of Graphs

2021 ◽  
Vol 54 ◽  
Author(s):  
Shariefuddin Pirzada ◽  
Saleem Khan
Keyword(s):  
Lower Bounds ◽  
Connected Graph ◽  
Distance Matrix ◽  
Laplacian Matrix ◽  
Wiener Index ◽  
Upper Bounds ◽  
Diagonal Matrix ◽  
Laplacian Eigenvalues ◽  
Eigenvalues Of Graphs

Let $G$ be a connected graph with $n$ vertices, $m$ edges and having diameter $d$. The distance Laplacian matrix $D^{L}$ is defined as $D^L=$Diag$(Tr)-D$, where Diag$(Tr)$ is the diagonal matrix of vertex transmissions and $D$ is the distance matrix of $G$. The distance Laplacian eigenvalues of $G$ are the eigenvalues of $D^{L}$ and are denoted by $\delta_{1}, ~\delta_{1},~\dots,\delta_{n}$. In this paper, we obtain (a) the upper bounds for the sum of $k$ largest and (b) the lower bounds for the sum of $k$ smallest non-zero, distance Laplacian eigenvalues of $G$ in terms of order $n$, diameter $d$ and Wiener index $W$ of $G$. We characterize the extremal cases of these bounds. As a consequence, we also obtain the bounds for the sum of the powers of the distance Laplacian eigenvalues of $G$.

scholarly journals WIENER INDEX OF TREES OF GIVEN ORDER AND DIAMETER AT MOST

2013 ◽  
Vol 89 (3) ◽  
pp. 379-396 ◽  
Author(s):  
SIMON MUKWEMBI ◽  
TOMÁŠ VETRÍK
Keyword(s):  
Upper Bound ◽  
Open Problem ◽  
Wiener Index ◽  
Upper Bounds ◽  
Degree Distance ◽  
Gutman Index

AbstractThe long-standing open problem of finding an upper bound for the Wiener index of a graph in terms of its order and diameter is addressed. Sharp upper bounds are presented for the Wiener index, and the related degree distance and Gutman index, for trees of order$n$and diameter at most$6$.

scholarly journals Wiener Index and Remoteness in Triangulations and Quadrangulations

2021 ◽  
Vol vol. 23 no. 1 (Graph Theory) ◽  
Author(s):  
Éva Czabarka ◽  
Peter Dankelmann ◽  
Trevor Olsen ◽  
László A. Székely
Keyword(s):  
Connected Graph ◽  
Wiener Index ◽  
Upper Bounds ◽  
Arithmetic Mean ◽  
Asymptotic Formulae

Let $G$ be a a connected graph. The Wiener index of a connected graph is the sum of the distances between all unordered pairs of vertices. We provide asymptotic formulae for the maximum Wiener index of simple triangulations and quadrangulations with given connectivity, as the order increases, and make conjectures for the extremal triangulations and quadrangulations based on computational evidence. If $\overline{\sigma}(v)$ denotes the arithmetic mean of the distances from $v$ to all other vertices of $G$, then the remoteness of $G$ is defined as the largest value of $\overline{\sigma}(v)$ over all vertices $v$ of $G$. We give sharp upper bounds on the remoteness of simple triangulations and quadrangulations of given order and connectivity.

scholarly journals Reciprocal product-degree distance of graphs

Filomat ◽  
2016 ◽  
Vol 30 (8) ◽  
pp. 2217-2231
Author(s):  
Guifu Su ◽  
Liming Xiong ◽  
Ivan Gutman ◽  
Lan Xu
Keyword(s):  
Connected Graph ◽  
Upper Bounds ◽  
Graph Invariant ◽  
Lower And Upper Bounds ◽  
Harary Index ◽  
Underlying Graph ◽  
Degree Distance ◽  
Degree Modification ◽  
Type Relation

We investigate a new graph invariant named reciprocal product-degree distance, defined as: RDD* = ?{u,v}?V(G)u?v deg(u)?deg(v)/dist(u,v) where deg(v) is the degree of the vertex v, and dist(u,v) is the distance between the vertices u and v in the underlying graph. RDD* is a product-degree modification of the Harary index. We determine the connected graph of given order with maximum RDD*-value, and establish lower and upper bounds for RDD*. Also a Nordhaus-Gaddum-type relation for RDD* is obtained.

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