scholarly journals The Terminal Hosoya Polynomial of Some Families of Composite Graphs

2014 ◽  
Vol 2014 ◽  
pp. 1-4 ◽  
Author(s):  
Emeric Deutsch ◽  
Juan Alberto Rodríguez-Velázquez
Keyword(s):  
Connected Graph ◽  
Product Graphs ◽  
Hosoya Polynomial ◽  
Pendent Vertices ◽  
Corona Product

Let G be a connected graph and let Ω(G) be the set of pendent vertices of G. The terminal Hosoya polynomial of G is defined as TH(G,t)∶=∑x,y∈Ω(G):x≠ytdG(x,y), where dG(x,y) denotes the distance between the pendent vertices x and y. In this note paper we obtain closed formulae for the terminal Hosoya polynomial of rooted product graphs and corona product graphs.

scholarly journals Zagreb Eccentricity Indices of the Generalized Hierarchical Product Graphs and Their Applications

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
Zhaoyang Luo ◽  
Jianliang Wu
Keyword(s):  
Connected Graph ◽  
Explicit Formulas ◽  
Connected Graphs ◽  
Product Graphs ◽  
Corona Product

LetGbe a connected graph. The first and second Zagreb eccentricity indices ofGare defined asM1*(G)=∑v∈V(G)‍εG2(v)andM2*(G)=∑uv∈E(G)‍εG(u)εG(v), whereεG(v)is the eccentricity of the vertexvinGandεG2(v)=(εG(v))2. Suppose thatG(U)⊓H(∅≠U⊆V(G))is the generalized hierarchical product of two connected graphsGandH. In this paper, the Zagreb eccentricity indicesM1*andM2*ofG(U)⊓Hare computed. Moreover, we present explicit formulas for theM1*andM2*ofS-sum graph, Cartesian, cluster, and corona product graphs by means of some invariants of the factors.

scholarly journals The metric dimension of strong product graphs

2015 ◽  
Vol 31 (2) ◽  
pp. 261-268
Author(s):  
JUAN A. RODRIGUEZ-VELAZQUEZ ◽  
◽  
DOROTA KUZIAK ◽  
ISMAEL G. YERO ◽  
JOSE M. SIGARRETA ◽  
...  
Keyword(s):  
Connected Graph ◽  
Metric Dimension ◽  
Np Hard ◽  
Strong Product ◽  
Product Graphs ◽  
Metric Basis

For an ordered subset S = {s1, s2, . . . sk} of vertices in a connected graph G, the metric representation of a vertex u with respect to the set S is the k-vector r(u|S) = (dG(v, s1), dG(v, s2), . . . , dG(v, sk)), where dG(x, y) represents the distance between the vertices x and y. The set S is a metric generator for G if every two different vertices of G have distinct metric representations with respect to S. A minimum metric generator is called a metric basis for G and its cardinality, dim(G), the metric dimension of G. It is well known that the problem of finding the metric dimension of a graph is NP-Hard. In this paper we obtain closed formulae and tight bounds for the metric dimension of strong product graphs.

scholarly journals Joins, coronas and their vertex-edge Wiener polynomials

2016 ◽  
Vol 47 (2) ◽  
pp. 163-178
Author(s):  
Mahdieh Azari ◽  
Ali Iranmanesh
Keyword(s):  
Generating Function ◽  
Connected Graph ◽  
Wiener Index ◽  
First Derivative ◽  
Sum Of Distances ◽  
Simple Connected Graph ◽  
Product Of Graphs ◽  
Corona Product

The vertex-edge Wiener index of a simple connected graph $G$ is defined as the sum of distances between vertices and edges of $G$. The vertex-edge Wiener polynomial of $G$ is a generating function whose first derivative is a $q-$analog of the vertex-edge Wiener index. Two possible distances $D_1(u, e|G)$ and $D_2(u, e|G)$ between a vertex $u$ and an edge $e$ of $G$ can be considered and corresponding to them, the first and second vertex-edge Wiener indices of $G$, and the first and second vertex-edge Wiener polynomials of $G$ are introduced. In this paper, we study the behavior of these indices and polynomials under the join and corona product of graphs. Results are applied for some classes of graphs such as suspensions, bottlenecks, and thorny graphs.

scholarly journals On the metric dimension of corona product graphs

2011 ◽  
Vol 61 (9) ◽  
pp. 2793-2798 ◽  
Author(s):  
I.G. Yero ◽  
D. Kuziak ◽  
J.A. Rodríguez-Velázquez
Keyword(s):  
Metric Dimension ◽  
Product Graphs ◽  
Corona Product

scholarly journals On the strong metric dimension of corona product graphs and join graphs

2013 ◽  
Vol 161 (7-8) ◽  
pp. 1022-1027 ◽  
Author(s):  
Dorota Kuziak ◽  
Ismael G. Yero ◽  
Juan A. Rodríguez-Velázquez
Keyword(s):  
Metric Dimension ◽  
Product Graphs ◽  
Strong Metric Dimension ◽  
Corona Product

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 Rainbow coloring in some corona product graphs

2019 ◽  
Vol 7 (1) ◽  
pp. 127-131 ◽  
Author(s):  
Kulkarni Sunita Jagannatharao ◽  
R. Murali
Keyword(s):  
Product Graphs ◽  
Corona Product ◽  
Rainbow Coloring

scholarly journals Analogies between the geodetic number and the Steiner number of some classes of graphs

Filomat ◽  
2015 ◽  
Vol 29 (8) ◽  
pp. 1781-1788 ◽  
Author(s):  
Ismael Yero ◽  
Juan Rodríguez-Velázquez
Keyword(s):  
Complete Graph ◽  
Open Problem ◽  
Minimum Size ◽  
Discrete Mathematics ◽  
Connected Subgraph ◽  
Minimum Cardinality ◽  
Geodetic Number ◽  
Product Graphs ◽  
Geodetic Set ◽  
Corona Product

A set of vertices S of a graph G is a geodetic set of G if every vertex v ? S lies on a shortest path between two vertices of S. The minimum cardinality of a geodetic set of G is the geodetic number of G and it is denoted by 1(G). A Steiner set of G is a set of vertices W of G such that every vertex of G belongs to the set of vertices of a connected subgraph of minimum size containing the vertices of W. The minimum cardinality of a Steiner set of G is the Steiner number of G and it is denoted by s(G). Let G and H be two graphs and let n be the order of G. The corona product G ? H is defined as the graph obtained from G and H by taking one copy of G and n copies of H and joining by an edge each vertex from the ith-copy of H to the ith-vertex of G. We study the geodetic number and the Steiner number of corona product graphs. We show that if G is a connected graph of order n ? 2 and H is a non complete graph, then g(G ? H) ? s(G ? H), which partially solve the open problem presented in [Discrete Mathematics 280 (2004) 259-263] related to characterize families of graphs G satisfying that g(G) ? s(G).

scholarly journals On the Local Metric Dimension of Corona Product Graphs

2015 ◽  
Vol 39 (S1) ◽  
pp. 157-173 ◽  
Author(s):  
Juan A. Rodríguez-Velázquez ◽  
Gabriel A. Barragán-Ramírez ◽  
Carlos García Gómez
Keyword(s):  
Metric Dimension ◽  
Product Graphs ◽  
Corona Product

ON LAPLACIAN SPECTRA OF SOME CORONA PRODUCT GRAPHS AND APPLICATIONS

2021 ◽  
Vol 10 (3) ◽  
pp. 1259-1271
Author(s):  
I.J. Gogoi ◽  
B. Phukan ◽  
A. Pegu ◽  
A. Bharali
Keyword(s):  
Laplacian Spectra ◽  
Product Graphs ◽  
Corona Product
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