scholarly journals Average connectivity of minimally 2-connected graphs and average edge-connectivity of minimally 2-edge-connected graphs

2021 ◽  
Vol 289 ◽  
pp. 233-247
Author(s):  
Rocío M. Casablanca ◽  
Lucas Mol ◽  
Ortrud R. Oellermann
Keyword(s):  
Edge Connectivity ◽  
Connected Graphs ◽  
Average Connectivity

scholarly journals Major n-connected graphs

1989 ◽  
Vol 47 (1) ◽  
pp. 43-52
Author(s):  
Ortrud R. Oellermann
Keyword(s):  
Positive Integer ◽  
Lower Bounds ◽  
Upper And Lower Bounds ◽  
Connected Subgraph ◽  
Edge Connectivity ◽  
Maximum Order ◽  
Induced Subgraph ◽  
Connected Graphs

AbstractAn induced subgraph H of connectivity (edge-connectivity) n in a graph G is a major n-connected (major n-edge-connected) subgraph of G if H contains no subgraph with connectivity (edge- connectivity) exceeding n and H has maximum order with respect to this property. An induced subgraph is a major (major edge-) subgraph if it is a major n-connected (major n-edge-connected) subgraph for some n. Let m be the maximum order among all major subgraphs of C. Then the major connectivity set K(G) of G is defined as the set of all n for which there exists a major n-connected subgraph of G having order m. The major edge-connectivity set is defined analogously. The connectivity and the elements of the major connectivity set of a graph are compared, as are the elements of the major connectivity set and the major edge-connectivity set of a graph. It is shown that every set S of nonnegative integers is the major connectivity set of some graph G. Further, it is shown that for each positive integer m exceeding every element of S, there exists a graph G such that every major k-connected subgraph of G, where k ∈ K(G), has order m. Moreover, upper and lower bounds on the order of such graphs G are established.

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 Average connectivity and average edge-connectivity in graphs

2013 ◽  
Vol 313 (20) ◽  
pp. 2232-2238 ◽  
Author(s):  
Jaehoon Kim ◽  
Suil O
Keyword(s):  
Edge Connectivity ◽  
Average Connectivity

scholarly journals Separation of Cartesian products of graphs into several connected components by the removal of edges

2021 ◽  
pp. 18-18
Author(s):  
Simon Spacapan
Keyword(s):  
Cartesian Product ◽  
Connected Components ◽  
Edge Connectivity ◽  
Cartesian Products ◽  
Minimum Cardinality ◽  
Connected Graphs ◽  
Cartesian Products Of Graphs

Let G = (V (G),E(G)) be a graph. A set S ? E(G) is an edge k-cut in G if the graph G-S = (V (G), E(G) \ S) has at least k connected components. The generalized k-edge connectivity of a graph G, denoted as ?k(G), is the minimum cardinality of an edge k-cut in G. In this article we determine generalized 3-edge connectivity of Cartesian product of connected graphs G and H and describe the structure of any minimum edge 3-cut in G2H. The generalized 3-edge connectivity ?3(G2H) is given in terms of ?3(G) and ?3(H) and in terms of other invariants of factors G and H.

scholarly journals A Characterization of Graphs by Codes from their Incidence Matrices

10.37236/2770 ◽  
2013 ◽  
Vol 20 (3) ◽  
Author(s):  
Peter Dankelmann ◽  
Jennifer D. Key ◽  
Bernardo G. Rodrigues
Keyword(s):  
Incidence Matrix ◽  
Linear Codes ◽  
Minimum Weight ◽  
Weight Enumerator ◽  
Edge Connectivity ◽  
Connected Graphs ◽  
Incidence Matrices ◽  
Wide Range ◽  
The Matrix

We continue our earlier investigation of properties of linear codes generated by the rows of incidence matrices of $k$-regular connected graphs on $n$ vertices. The notion of edge connectivity is used to show that, for a wide range of such graphs, the $p$-ary code, for all primes $p$, from an $n \times \frac{1}{2}nk$ incidence matrix has dimension $n$ or $n-1$, minimum weight $k$, the minimum words are the scalar multiples of the rows, there is a gap in the weight enumerator between $k$ and $2k-2$, and the words of weight $2k-2$ are the scalar multiples of the differences of intersecting rows of the matrix. For such graphs, the graph can thus be retrieved from the code.

scholarly journals Extremal topological indices for graphs of given connectivity

Filomat ◽  
2015 ◽  
Vol 29 (7) ◽  
pp. 1639-1643 ◽  
Author(s):  
Ioan Tomescu ◽  
Misbah Arshad ◽  
Muhammad Jamil
Keyword(s):  
Wiener Index ◽  
Topological Indices ◽  
Connectivity Index ◽  
Zeroth Order ◽  
Edge Connectivity ◽  
Connected Graphs ◽  
Randic Index ◽  
Randić Connectivity Index ◽  
Unique Graph ◽  
General Randić Connectivity Index

In this paper, we show that in the class of graphs of order n and given (vertex or edge) connectivity equal to k (or at most equal to k), 1 ? k ? n - 1, the graph Kk + (K1? Kn-k-1) is the unique graph such that zeroth-order general Randic index, general sum-connectivity index and general Randic connectivity index are maximum and general hyper-Wiener index is minimum provided ? > 1. Also, for 2-connected (or 2-edge connected) graphs and ? > 0 the unique graph minimizing these indices is the n-vertex cycle Cn.

scholarly journals The Formula to Count The Number of Vertices Labeled Order Six Connected Graphs with Maximum Thirty Edges without Loops

2021 ◽  
Vol 1751 ◽  
pp. 012023
Author(s):  
F C Puri ◽  
Wamiliana ◽  
M Usman ◽  
Amanto ◽  
M Ansori ◽  
...  
Keyword(s):  
Connected Graphs

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.

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