scholarly journals A note on the edge reconstruction conjecture

1975 ◽  
Vol 12 (1) ◽  
pp. 27-30 ◽  
Author(s):  
E.F. Schmeichel
Keyword(s):  
Degree Sequence ◽  
Existence Condition ◽  
Vertex Degree ◽  
Hamiltonian Cycles ◽  
Edge Reconstruction

Let G be a graph with vertex degree sequence d1 ≤ d2 ≤ … ≤ dp It is shown that if di + dp–i+1 ≥ p for some i, then G is uniquely reconstructable from its collection of maximal (edge deleted) subgraphs. This generalizes considerably a result of Lovász. As a corollary, it is shown that Chvátal's existence condition for hamiltonian cycles implies edge reconstructability as well.

scholarly journals On relations between Kirchhoff index, Laplacian energy, Laplacian-energy-like invariant and degree deviation of graphs

Filomat ◽  
2020 ◽  
Vol 34 (3) ◽  
pp. 1025-1033
Author(s):  
Predrag Milosevic ◽  
Emina Milovanovic ◽  
Marjan Matejic ◽  
Igor Milovanovic
Keyword(s):  
Topological Index ◽  
Degree Sequence ◽  
Laplacian Matrix ◽  
Vertex Degree ◽  
Graph Invariants ◽  
Kirchhoff Index ◽  
Laplacian Eigenvalues ◽  
Laplacian Energy ◽  
Degree Deviation ◽  
Simple Connected Graph

Let G be a simple connected graph of order n and size m, vertex degree sequence d1 ? d2 ?...? dn > 0, and let ?1 ? ? 2 ? ... ? ?n-1 > ?n = 0 be the eigenvalues of its Laplacian matrix. Laplacian energy LE, Laplacian-energy-like invariant LEL and Kirchhoff index Kf, are graph invariants defined in terms of Laplacian eigenvalues. These are, respectively, defined as LE(G) = ?n,i=1 |?i-2m/n|, LEL(G) = ?n-1 i=1 ??i and Kf (G) = n ?n-1,i=1 1/?i. A vertex-degree-based topological index referred to as degree deviation is defined as S(G) = ?n,i=1 |di- 2m/n|. Relations between Kf and LE, Kf and LEL, as well as Kf and S are obtained.

A Small-World Model of Scale-Free Networks: Features and Verifications

2011 ◽  
Vol 50-51 ◽  
pp. 166-170 ◽  
Author(s):  
Wen Jun Xiao ◽  
Shi Zhong Jiang ◽  
Guan Rong Chen
Keyword(s):  
Complex Networks ◽  
Power Law ◽  
Degree Sequence ◽  
Small World ◽  
Static Condition ◽  
Vertex Degree ◽  
Power Law Distribution ◽  
Least Common Multiple ◽  
Scale Free ◽  
Vertex Degrees

It is now well known that many large-sized complex networks obey a scale-free power-law vertex-degree distribution. Here, we show that when the vertex degrees of a large-sized network follow a scale-free power-law distribution with exponent  2, the number of degree-1 vertices, if nonzero, is of order N and the average degree is of order lower than log N, where N is the size of the network. Furthermore, we show that the number of degree-1 vertices is divisible by the least common multiple of , , . . ., , and l is less than log N, where l = < is the vertex-degree sequence of the network. The method we developed here relies only on a static condition, which can be easily verified, and we have verified it by a large number of real complex networks.

scholarly journals Degree Sequence of Graph Operator for some Standard Graphs

2020 ◽  
Vol 5 (2) ◽  
pp. 99-108
Author(s):  
◽  
P. S Ranjini ◽  
V. Lokesha ◽  
Sandeep Kumar
Keyword(s):  
Degree Sequence ◽  
Topological Indices ◽  
Vertex Degree ◽  
Mathematical Chemistry

AbstractTopological indices play a very important role in the mathematical chemistry. The topological indices are numerical parameters of a graph. The degree sequence is obtained by considering the set of vertex degree of a graph. Graph operators are the ones which are used to obtain another broader graphs. This paper attempts to find degree sequence of vertex–F join operation of graphs for some standard graphs.

scholarly journals On some mathematical properties of the general zeroth-order Randić coindex of graphs

2020 ◽  
Vol 12 (2) ◽  
pp. 75-82
Author(s):  
I. Milovanović ◽  
M. Matejić ◽  
E. Milovanović ◽  
A. Ali
Keyword(s):  
Real Number ◽  
Lower Bounds ◽  
Connected Graph ◽  
Degree Sequence ◽  
Tree Structure ◽  
Vertex Degree ◽  
Arbitrary Real Number ◽  
Zeroth Order ◽  
Mathematical Properties ◽  
Simple Connected Graph

Let G = (V,E), V = {v1, v2,..., vn}, be a simple connected graph of order n, size m with vertex degree sequence ∆ = d1 ≥ d2 ≥ ··· ≥ dn = d > 0, di = d(vi). Denote by G a complement of G. If vertices vi and v j are adjacent in G, we write i ~ j, otherwise we write i j. The general zeroth-order Randic coindex of ' G is defined as 0Ra(G) = ∑i j (d a-1 i + d a-1 j ) = ∑ n i=1 (n-1-di)d a-1 i , where a is an arbitrary real number. Similarly, general zerothorder Randic coindex of ' G is defined as 0Ra(G) = ∑ n i=1 di(n-1-di) a-1 . New lower bounds for 0Ra(G) and 0Ra(G) are obtained. A case when G has a tree structure is also covered.

scholarly journals Minimum vertex degree condition for tight Hamiltonian cycles in 3‐uniform hypergraphs

2019 ◽  
Vol 119 (2) ◽  
pp. 409-439 ◽  
Author(s):  
Christian Reiher ◽  
Vojtěch Rödl ◽  
Andrzej Ruciński ◽  
Mathias Schacht ◽  
Endre Szemerédi
Keyword(s):  
Vertex Degree ◽  
Hamiltonian Cycles ◽  
Degree Condition ◽  
Uniform Hypergraphs

Creating recreational Hamiltonian cycle problems

2004 ◽  
Vol 88 (512) ◽  
pp. 215-218 ◽  
Author(s):  
Mark A. M. Lynch
Keyword(s):  
Hamiltonian Cycle ◽  
Vertex Degree ◽  
Hamiltonian Cycles ◽  
Arbitrary Size ◽  
Method Of Solution ◽  
Average Vertex

In this paper graphs that contain unique Hamiltonian cycles are introduced. The graphs are of arbitrary size and dense in the sense that their average vertex degree is greater than half the number of vertices that make up the graph. The graphs can be used to generate challenging puzzles. The problem is particularly challenging when the graph is large and the ‘method’ of solution is unknown to the solver.

scholarly journals On the Randić Index of Corona Product Graphs

2011 ◽  
Vol 2011 ◽  
pp. 1-7
Author(s):  
Ismael G. Yero ◽  
Juan A. Rodríguez-Velázquez
Keyword(s):  
Degree Sequence ◽  
Upper Bounds ◽  
Vertex Degree ◽  
Regular Graphs ◽  
Randić Index ◽  
Lower And Upper Bounds ◽  
Randic Index ◽  
Product Graphs ◽  
Vertex Set ◽  
Corona Product

Let G be a graph with vertex set V=(v1,v2,…,vn). Let δ(vi) be the degree of the vertex vi∈V. If the vertices vi1,vi2,…,vih+1 form a path of length h≥1 in the graph G, then the hth order Randić index Rh of G is defined as the sum of the terms 1/δ(vi1)δ(vi2)⋯δ(vih+1) over all paths of length h contained (as subgraphs) in G. Lower and upper bounds for Rh, in terms of the vertex degree sequence of its factors, are obtained for corona product graphs. Moreover, closed formulas are obtained when the factors are regular graphs.

scholarly journals Resilient Degree Sequences with respect to Hamilton Cycles and Matchings in Random Graphs

10.37236/8279 ◽  
2019 ◽  
Vol 26 (4) ◽  
Author(s):  
Padraig Condon ◽  
Alberto Espuny Díaz ◽  
Daniela Kühn ◽  
Deryk Osthus ◽  
Jaehoon Kim
Keyword(s):  
Random Graphs ◽  
Perfect Matching ◽  
Degree Sequence ◽  
Hamilton Cycle ◽  
Vertex Degree ◽  
Degree Sequences ◽  
Hamilton Cycles ◽  
Degree Condition ◽  
Degree Conditions

Pósa's theorem states that any graph $G$ whose degree sequence $d_1 \le \cdots \le d_n$ satisfies $d_i \ge i+1$ for all $i < n/2$ has a Hamilton cycle. This degree condition is best possible. We show that a similar result holds for suitable subgraphs $G$ of random graphs, i.e.~we prove a `resilience version' of Pósa's theorem: if $pn \ge C \log n$ and the $i$-th vertex degree (ordered increasingly) of $G \subseteq G_{n,p}$ is at least $(i+o(n))p$ for all $i<n/2$, then $G$ has a Hamilton cycle. This is essentially best possible and strengthens a resilience version of Dirac's theorem obtained by Lee and Sudakov. Chvátal's theorem generalises Pósa's theorem and characterises all degree sequences which ensure the existence of a Hamilton cycle. We show that a natural guess for a resilience version of Chvátal's theorem fails to be true. We formulate a conjecture which would repair this guess, and show that the corresponding degree conditions ensure the existence of a perfect matching in any subgraph of $G_{n,p}$ which satisfies these conditions. This provides an asymptotic characterisation of all degree sequences which resiliently guarantee the existence of a perfect matching.

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