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.

ON EDGE-CONNECTIVITY AND SUPER EDGE-CONNECTIVITY OF INTERCONNECTION NETWORKS MODELED BY PRODUCT GRAPHS

2010 ◽  
Vol 02 (02) ◽  
pp. 143-150
Author(s):  
CHUNXIANG WANG
Keyword(s):  
Connected Graph ◽  
Interconnection Networks ◽  
Minimum Degree ◽  
Edge Connectivity ◽  
Minimum Cardinality ◽  
Connected Graphs ◽  
Product Graph ◽  
Product Graphs

The super edge-connectivity λ′ of a connected graph G is the minimum cardinality of an edge-cut F in G such that every component of G–F contains at least two vertices. Let two connected graphs Gm and Gp have m and p vertices, minimum degree δ(Gm) and δ(Gp), edge-connectivity λ(Gm) and λ(Gp), respectively. This paper shows that min {pλ(Gm), λ(Gp) + δ(Gm), δ(Gm)(λ(Gp) + 1), (δ(Gm) + 1)λ(Gp)} ≤ λ(Gm * Gp) ≤ δ(Gm) + δ(Gp), where the product graph Gm * Gp of two given graphs Gm and Gp, defined by J. C. Bermond et al. [J. Combin. Theory B36 (1984) 32–48] in the context of the so-called (△, D)-problem, is one interesting model in the design of large reliable networks. Moreover, this paper determines λ′(Gm * Gp) ≤ min {pδ(Gm), ξ(Gp) + 2δ(Gm)} and λ′(G1 ⊕ G2) ≥ min {n, λ1 + λ2} if δ1 = δ2.

scholarly journals Fractional Local Metric Dimension of Comb Product Graphs

2020 ◽  
Vol 17 (4) ◽  
pp. 1288
Author(s):  
Siti Aisyah ◽  
Mohammad Imam Utoyo ◽  
Liliek Susilowati
Keyword(s):  
Connected Graph ◽  
Metric Dimension ◽  
Connected Graphs ◽  
Product Graphs

The local resolving neighborhood  of a pair of vertices  for  and  is if there is a vertex  in a connected graph  where the distance from  to  is not equal to the distance from  to , or defined by . A local resolving function  of  is a real valued function   such that  for  and . The local fractional metric dimension of graph  denoted by , defined by  In this research, the author discusses about the local fractional metric dimension of comb product are two graphs, namely graph  and graph , where graph  is a connected graphs and graph  is a complate graph  and denoted by  We get

Applications on Hyper-Zagreb Index of Generalized Hierarchical Product Graphs

2016 ◽  
Vol 13 (10) ◽  
pp. 7355-7361 ◽  
Author(s):  
Zhaoyang Luo
Keyword(s):  
Connected Graph ◽  
Explicit Formulas ◽  
Zagreb Index ◽  
Hexagonal Chain ◽  
Product Graphs

Let G be a connected graph. The Hyper-Zagreb index of a connected graph G is defined as HM(G) = Σuv∈EG [dG(u)+dG(v)]2, where dG(v) is the degree of the vertex v in G. In this paper, the Hyper-Zagreb Gindex of the generalized hierarchical, Cartesian, cluster, corona products and four new sums of graphs according to some invariants of the factors are computed, respectively. As applications, we present explicit formulas for the HM index of the linear phenylene Fn, the C4 nanotorus Cm□Cn, the C4 nanotubes Pm□Cn, the l-dimensional hypercubes Ql , the zig-zag polyhex nanotube TUHC6[2n, 2], the hexagonal chain ln, the regular dicentric dendrimer DDp,r and so forth.

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 Graph weights arising from Mayer and Ree-Hoover theories of virial expansions

2008 ◽  
Vol DMTCS Proceedings vol. AJ,... (Proceedings) ◽  
Author(s):  
Amel Kaouche ◽  
Pierre Leroux
Keyword(s):  
Connected Graph ◽  
Convex Polytopes ◽  
Hard Core ◽  
Ideal Gas ◽  
One Dimension ◽  
Graph Invariants ◽  
Explicit Formulas ◽  
Connected Graphs ◽  
International Audience

International audience We study graph weights (i.e., graph invariants) which arise naturally in Mayer's theory and Ree-Hoover's theory of virial expansions in the context of a non-ideal gas. We give special attention to the Second Mayer weight $w_M(c)$ and the Ree-Hoover weight $w_{RH}(c)$ of a $2$-connected graph $c$ which arise from the hard-core continuum gas in one dimension. These weights are computed using signed volumes of convex polytopes naturally associated with the graph $c$. Among our results are the values of Mayer's weight and Ree-Hoover's weight for all $2$-connected graphs $b$ of size at most $8$, and explicit formulas for certain infinite families. Nous étudions les poids de graphes (c'est-à-dire, les invariants de graphes) qui apparaissent naturellement dans la théorie de Mayer et la théorie de Ree-Hoover pour le développement du viriel dans le contexte d'un gaz imparfait. Nous donnons une attention particulière au deuxième poids $w_M(c)$ de Mayer et au poids $w_{RH}(c)$ de Ree-Hoover d'un graphe $2$-connexe $c$ dans le cas d'un gaz à noyaux durs et à positions continues en une dimension. Ces poids sont calculés à partir de volumes signés de polytopes convexes associés naturellement au graphe $c$. Parmi nos résultats sont les valeurs du poids de Mayer et du poids de Ree-Hoover pour tous les graphes $2$-connexes $b$ de taille au plus $8$, et des formules explicites pour certaines familles infinies.

scholarly journals On minimum algebraic connectivity of graphs whose complements are bicyclic

2019 ◽  
Vol 17 (1) ◽  
pp. 1490-1502 ◽  
Author(s):  
Jia-Bao Liu ◽  
Muhammad Javaid ◽  
Mohsin Raza ◽  
Naeem Saleem
Keyword(s):  
Connected Graph ◽  
Diffusion Processes ◽  
Laplacian Matrix ◽  
Group Differences ◽  
Algebraic Connectivity ◽  
Connected Graphs ◽  
Bicyclic Graph ◽  
Smallest Eigenvalue ◽  
Unique Graph ◽  
The Stability

Abstract The second smallest eigenvalue of the Laplacian matrix of a graph (network) is called its algebraic connectivity which is used to diagnose Alzheimer’s disease, distinguish the group differences, measure the robustness, construct multiplex model, synchronize the stability, analyze the diffusion processes and find the connectivity of the graphs (networks). A connected graph containing two or three cycles is called a bicyclic graph if its number of edges is equal to its number of vertices plus one. In this paper, firstly the unique graph with a minimum algebraic connectivity is characterized in the class of connected graphs whose complements are bicyclic with exactly three cycles. Then, we find the unique graph of minimum algebraic connectivity in the class of connected graphs $\begin{array}{} {\it\Omega}^c_{n}={\it\Omega}^c_{1,n}\cup{\it\Omega}^c_{2,n}, \end{array}$ where $\begin{array}{} {\it\Omega}^c_{1,n} \end{array}$ and $\begin{array}{} {\it\Omega}^c_{2,n} \end{array}$ are classes of the connected graphs in which the complement of each graph of order n is a bicyclic graph with exactly two and three cycles, respectively.

A NOTE ON GENERALIZED RAINBOW CONNECTION OF CONNECTED GRAPHS AND THEIR NUMBER OF EDGES

2021 ◽  
Vol 66 (3) ◽  
pp. 3-7
Author(s):  
Anh Nguyen Thi Thuy ◽  
Duyen Le Thi
Keyword(s):  
Upper Bound ◽  
Connected Graph ◽  
Connection Number ◽  
Connected Graphs ◽  
Rainbow Connection Number ◽  
Rainbow Connection ◽  
Rainbow Path ◽  
Vertex Disjoint

Let l ≥ 1, k ≥ 1 be two integers. Given an edge-coloured connected graph G. A path P in the graph G is called l-rainbow path if each subpath of length at most l + 1 is rainbow. The graph G is called (k, l)-rainbow connected if any two vertices in G are connected by at least k pairwise internally vertex-disjoint l-rainbow paths. The smallest number of colours needed in order to make G (k, l)-rainbow connected is called the (k, l)-rainbow connection number of G and denoted by rck,l(G). In this paper, we first focus to improve the upper bound of the (1, l)-rainbow connection number depending on the size of connected graphs. Using this result, we characterize all connected graphs having the large (1, 2)-rainbow connection number. Moreover, we also determine the (1, l)-rainbow connection number in a connected graph G containing a sequence of cut-edges.

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 The Structure of Locally Finite Two-Connected Graphs

10.37236/1211 ◽  
1995 ◽  
Vol 2 (1) ◽  
Author(s):  
Carl Droms ◽  
Brigitte Servatius ◽  
Herman Servatius
Keyword(s):  
Connected Graph ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Locally Finite ◽  
Connected Graphs ◽  
Finite Graphs ◽  
Locally Finite Graphs ◽  
Necessary And Sufficient ◽  
Block Tree ◽  
Labeled Tree

We expand on Tutte's theory of $3$-blocks for $2$-connected graphs, generalizing it to apply to infinite, locally finite graphs, and giving necessary and sufficient conditions for a labeled tree to be the $3$-block tree of a $2$-connected graph.

scholarly journals On generalized 3-connectivity of the strong product of graphs

2018 ◽  
Vol 12 (2) ◽  
pp. 297-317
Author(s):  
Encarnación Abajo ◽  
Rocío Casablanca ◽  
Ana Diánez ◽  
Pedro García-Vázquez
Keyword(s):  
Positive Integer ◽  
Connected Graph ◽  
Sharp Bound ◽  
Connected Graphs ◽  
Strong Product ◽  
Generalized Connectivity ◽  
Internally Disjoint Trees ◽  
Product Of Graphs

Let G be a connected graph with n vertices and let k be an integer such that 2 ? k ? n. The generalized connectivity kk(G) of G is the greatest positive integer l for which G contains at least l internally disjoint trees connecting S for any set S ? V (G) of k vertices. We focus on the generalized connectivity of the strong product G1 _ G2 of connected graphs G1 and G2 with at least three vertices and girth at least five, and we prove the sharp bound k3(G1 _ G2) ? k3(G1)_3(G2) + k3(G1) + k3(G2)-1.

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