Trees and Tree-Equivalent Graphs

1965 ◽  
Vol 17 ◽  
pp. 731-733 ◽  
Author(s):  
C. Ramanujacharyulu
Keyword(s):  
Connected Graph ◽  
Connected Components ◽  
Cyclomatic Number ◽  
Equivalent Graph ◽  
Theory Of Graphs

As is well known in the theory of graphs a tree is a connected graph without cycles. Many characterizing properties of trees are known (1), for example the cyclomatic number is equal to zero, which is also equal to p — 1, where p is the number of connected components of the graph. The graphs with cyclomatic number equal to p — 1 are defined here as tree-equivalent graphs. A tree is always a tree-equivalent graph but not conversely. The properties of tree-equivalent graphs are studied here.

The Maximum Genus of Cartesian Products of Graphs

1974 ◽  
Vol 26 (5) ◽  
pp. 1025-1035 ◽  
Author(s):  
Joseph Zaks
Keyword(s):  
Connected Graph ◽  
Connected Components ◽  
Open Disk ◽  
Maximum Genus ◽  
Cartesian Products ◽  
Euler’S Formula ◽  
Cartesian Products Of Graphs

The maximum genus γM(G) of a connected graph G has been defined in [2] as the maximum g for which there exists an embedding h : G —> S(g), where S(g) is a compact orientable 2-manifold of genus g, such that each one of the connected components of S(g) — h(G) is homeomorphic to an open disk; such an embedding is called cellular. If G is cellularly embedded in S(g), having V vertices, E edges and F faces, then by Euler's formulaV-E + F = 2-2g.

LOCAL GROUPS IN FREE GROUPOIDS SATISFYING CERTAIN MONOID IDENTITIES

2002 ◽  
Vol 12 (01n02) ◽  
pp. 357-369
Author(s):  
ALAIR PEREIRA DO LAGO
Keyword(s):  
Free Group ◽  
Connected Graph ◽  
Generating Set ◽  
Burnside Groups ◽  
Cyclomatic Number ◽  
Strongly Connected

Let G be a (possibly infinite) strongly connected graph and let [Formula: see text] be a set of monoid identities such that any monoid satisfying [Formula: see text] is also a group. Let ℬ be the free groupoid on G satisfying [Formula: see text]. Then, the local groups ℬv, for v ∈ V (G), are all isomorphic to a free group satisfying [Formula: see text]. Furthermore, it is free over a generating set which can be effectively characterized and whose cardinality is the cyclomatic number of the graph G. We also show applications that establish an important connection between free Burnside groups and free Burnside semigroups.

scholarly journals Undirecting membership in models of Anti-Foundation

2020 ◽  
Author(s):  
Bea Adam-Day ◽  
Peter J. Cameron
Keyword(s):  
Set Theory ◽  
Random Graph ◽  
Connected Graph ◽  
Countable Model ◽  
Connected Components ◽  
Membership Relation ◽  
Multiple Edges

AbstractIt is known that, if we take a countable model of Zermelo–Fraenkel set theory ZFC and “undirect” the membership relation (that is, make a graph by joining x to y if either $$x\in y$$ x ∈ y or $$y\in x$$ y ∈ x ), we obtain the Erdős–Rényi random graph. The crucial axiom in the proof of this is the Axiom of Foundation; so it is natural to wonder what happens if we delete this axiom, or replace it by an alternative (such as Aczel’s Anti-Foundation Axiom). The resulting graph may fail to be simple; it may have loops (if $$x\in x$$ x ∈ x for some x) or multiple edges (if $$x\in y$$ x ∈ y and $$y\in x$$ y ∈ x for some distinct x, y). We show that, in ZFA, if we keep the loops and ignore the multiple edges, we obtain the “random loopy graph” (which is $$\aleph _0$$ ℵ 0 -categorical and homogeneous), but if we keep multiple edges, the resulting graph is not $$\aleph _0$$ ℵ 0 -categorical, but has infinitely many 1-types. Moreover, if we keep only loops and double edges and discard single edges, the resulting graph contains countably many connected components isomorphic to any given finite connected graph with loops.

A combinatorial proof of a theorem of Tutte

1966 ◽  
Vol 62 (4) ◽  
pp. 683-684 ◽  
Author(s):  
John H. Halton
Keyword(s):  
Connected Components ◽  
Combinatorial Proof ◽  
Linear Graph ◽  
Theory Of Graphs

We refer to a beautiful and important result of Tutte(1), in the theory of graphs; that a linear graph G is prime if and only if it contains a set ∑ of vertices, such that u(G∑) > n(∑); where n(∑) is the number of vertices in ∑, G∑ is the graph obtained from G by deleting the star of ∑ (all the vertices of G in ∑, together with all the edges of G meeting vertices of ∑), and u(G∑) is the number of connected components of G∑ having an odd number of vertices.

Neighbor Rupture Degree of Transformation Graphs Gxy−

2017 ◽  
Vol 28 (04) ◽  
pp. 335-355
Author(s):  
Goksen Bacak-Turan ◽  
Ekrem Oz
Keyword(s):  
Connected Graph ◽  
Bipartite Graphs ◽  
Connected Components ◽  
Complete Graphs ◽  
Maximum Order ◽  
Wheel Graphs ◽  
Complete Bipartite Graphs

A vulnerability parameter the neighbor rupture degree can be used to obtain the vulnerability of a spy network. The neighbor rupture degree of a noncomplete connected graph [Formula: see text] is defined to be [Formula: see text] where [Formula: see text] is any vertex subversion strategy of [Formula: see text], [Formula: see text] is the number of connected components in [Formula: see text], and [Formula: see text] is the maximum order of the components of [Formula: see text]. In this study, the neighbor rupture degree of transformation graphs [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text] of path graphs, cycle graphs, wheel graphs, complete graphs and complete bipartite graphs are obtained.

scholarly journals ALGEBRAIC CHARACTERIZATIONS OF GRAPH IMBEDDABILITY IN SURFACES AND PSEUDOSURFACES

2006 ◽  
Vol 15 (06) ◽  
pp. 681-693 ◽  
Author(s):  
LOWELL ABRAMS ◽  
DANIEL C. SLILATY
Keyword(s):  
Connected Graph ◽  
Homology Group ◽  
Dual Graph ◽  
Connected Components ◽  
Homology Groups ◽  
The Given

Given a finite connected graph G and specifications for a closed, connected pseudosurface, we characterize when G can be imbedded in a closed, connected pseudosurface with the given specifications. The specifications for the pseudosurface are: the number of face-connected components, the number of pinches, the number of crosscaps and handles, and the dimension of the first ℤ2 homology group. The characterizations are formulated in terms of the existence of a dual graph G* on the same set of edges as G which satisfies algebraic conditions inspired by homology groups and their intersection products.

scholarly journals Computing Exact Values for Gutman Indices of Sum Graphs under Cartesian Product

2021 ◽  
Vol 2021 ◽  
pp. 1-20
Author(s):  
Abdulaziz Mohammed Alanazi ◽  
Faiz Farid ◽  
Muhammad Javaid ◽  
Augustine Munagi
Keyword(s):  
Connected Graph ◽  
Topological Index ◽  
Cartesian Product ◽  
Chemical Compounds ◽  
Factor Graphs ◽  
Cartesian Products ◽  
Degree Distance ◽  
Gutman Index ◽  
Extremal Theory ◽  
Theory Of Graphs

Gutman index of a connected graph is a degree-distance-based topological index. In extremal theory of graphs, there is great interest in computing such indices because of their importance in correlating the properties of several chemical compounds. In this paper, we compute the exact formulae of the Gutman indices for the four sum graphs (S-sum, R-sum, Q-sum, and T-sum) in the terms of various indices of their factor graphs, where sum graphs are obtained under the subdivision operations and Cartesian products of graphs. We also provide specific examples of our results and draw a comparison with previously known bounds for the four sum graphs.

scholarly journals Connectedness criteria for graphs by means of omega invariant

Filomat ◽  
2020 ◽  
Vol 34 (2) ◽  
pp. 647-652
Author(s):  
Utkum Sanli ◽  
Feriha Celik ◽  
Sadik Delen ◽  
Ismail Cangul
Keyword(s):  
Graph Theory ◽  
Euler Characteristic ◽  
Connected Graph ◽  
Topological Property ◽  
Degree Sequence ◽  
Direct Information ◽  
Combinatorial Properties ◽  
Cyclomatic Number ◽  
The Common ◽  
Disconnected Graph

A realizable degree sequence can be realized in many ways as a graph. There are several tests for determining realizability of a degree sequence. Up to now, not much was known about the common properties of these realizations. Euler characteristic is a well-known characteristic of graphs and their underlying surfaces. It is used to determine several combinatorial properties of a surface and of all graphs embedded onto it. Recently, last two authors defined a number ? which is invariant for all realizations of a given degree sequence. ? is shown to be related to Euler characteristic and cyclomatic number. Several properties of ? are obtained and some applications in extremal graph theory are done by authors. As already shown, the number gives direct information compared with the Euler characteristic on the realizability, number of realizations, being acyclic or cyclic, number of components, chords, loops, pendant edges, faces, bridges etc. In this paper, another important topological property of graphs which is connectedness is studied by means of ?. It is shown that all graphs with ?(G)?-4 are disconnected, and if ?(G)? -2, then the graph could be connected or disconnected. It is also shown that if the realization is a connected graph and ?(G)=-2, then certainly the graph should be acyclic. Similarly, it is shown that if the realization is a connected graph G and ?(G)? 0, then certainly the graph should be cyclic. Also, the fact that when ?(G)?-4, the components of the disconnected graph could not all be cyclic, and that if all the components of a graph G are cyclic, then ?(G) ? 0 are proven.

Neighbor Rupture Degree of Harary Graphs

2016 ◽  
Vol 27 (06) ◽  
pp. 739-756
Author(s):  
Ferhan Nihan Altundag ◽  
Goksen Bacak-Turan
Keyword(s):  
Connected Graph ◽  
Connected Components ◽  
Maximum Order ◽  
Communication Links

The vulnerability shows the endurance of the network until the communication collapse after the breakdown of certain stations or communication links. If a spy or a station is invaded in a spy network, then the adjacent stations are treacherous. A vulnerability parameter the neighbor rupture degree can be used to obtain the vulnerability of a spy network. The neighbor rupture degree of a noncomplete connected graph G is defined to be [Formula: see text] where S is any vertex subversion strategy of G, w(G/S) is the number of connected components in G/S, and c(G/S) is the maximum order of the components of G/S. In this paper, the neighbor rupture degree of Harary graphs are obtained.

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