scholarly journals On the General Position Number of Complementary Prisms

2021 ◽  
Vol 178 (3) ◽  
pp. 267-281
Author(s):  
P. K. Neethu ◽  
S.V. Ullas Chandran ◽  
Manoj Changat ◽  
Sandi Klavžar
Keyword(s):  
Lower Bound ◽  
Bipartite Graph ◽  
General Position ◽  
Disjoint Union ◽  
Perfect Matching ◽  
Split Graph ◽  
Block Graphs ◽  
Disconnected Graphs ◽  
Complementary Prism ◽  
Sharp Lower Bound

The general position number gp(G) of a graph G is the cardinality of a largest set of vertices S such that no element of S lies on a geodesic between two other elements of S. The complementary prism G G ¯ of G is the graph formed from the disjoint union of G and its complement G ¯ by adding the edges of a perfect matching between them. It is proved that gp(G G ¯ ) ≤ n(G) + 1 if G is connected and gp(G G ¯ ) ≤ n(G) if G is disconnected. Graphs G for which gp(G G ¯ ) = n(G) + 1 holds, provided that both G and G ¯ are connected, are characterized. A sharp lower bound on gp(G G ¯ ) is proved. If G is a connected bipartite graph or a split graph then gp(G G ¯ ) ∈ {n(G), n(G)+1}. Connected bipartite graphs and block graphs for which gp(G G ¯ ) = n(G) + 1 holds are characterized. A family of block graphs is constructed in which the gp-number of their complementary prisms is arbitrary smaller than their order.

scholarly journals On the geodetic hull number for complementary prisms II

2020 ◽  
Author(s):  
Diane Castonguay ◽  
Erika Morais Martins Coelho ◽  
Hebert Coelho ◽  
Julliano Nascimento
Keyword(s):  
Convex Hull ◽  
Shortest Path ◽  
Disjoint Union ◽  
Perfect Matching ◽  
Convex Set ◽  
Upper Bounds ◽  
The Other ◽  
Lower And Upper Bounds ◽  
Split Graph ◽  
Complementary Prism

In the geodetic convexity, a set of vertices $S$ of a graph $G$ is \textit{convex} if all vertices belonging to any shortest path between two vertices of $S$ lie in $S$. The \textit{convex hull} $H(S)$ of $S$ is the smallest convex set containing $S$. If $H(S) = V(G)$, then $S$ is a \textit{hull set}. The cardinality $h(G)$ of a minimum hull set of $G$ is the \textit{hull number} of $G$. The \textit{complementary prism} $G\overline{G}$ of a graph $G$ arises from the disjoint union of the graph $G$ and $\overline{G}$ by adding the edges of a perfect matching between the corresponding vertices of $G$ and $\overline{G}$. A graph $G$ is \textit{autoconnected} if both $G$ and $\overline{G}$ are connected. Motivated by previous work, we study the hull number for complementary prisms of autoconnected graphs. When $G$ is a split graph, we present lower and upper bounds showing that the hull number is unlimited. In the other case, when $G$ is a non-split graph, it is limited by $3$.

scholarly journals A sharp lower bound on the signless Laplacian index of graphs with (κ,τ)-regular sets

2018 ◽  
Vol 6 (1) ◽  
pp. 68-76 ◽  
Author(s):  
Milica Andeelić ◽  
Domingos M. Cardoso ◽  
António Pereira
Keyword(s):  
Lower Bound ◽  
Perfect Matching ◽  
Computational Experiments ◽  
Sufficient Condition ◽  
Hamiltonian Graphs ◽  
Laplacian Spectra ◽  
Signless Laplacian ◽  
Regular Sets ◽  
Sharp Lower Bound ◽  
Better Than

Abstract A new lower bound on the largest eigenvalue of the signless Laplacian spectra for graphs with at least one (κ,τ)regular set is introduced and applied to the recognition of non-Hamiltonian graphs or graphs without a perfect matching. Furthermore, computational experiments revealed that the introduced lower bound is better than the known ones. The paper also gives sufficient condition for a graph to be non Hamiltonian (or without a perfect matching).

scholarly journals A sharp lower bound for the least eigenvalue of the signless Laplacian of a non-bipartite graph

2008 ◽  
Vol 429 (11-12) ◽  
pp. 2770-2780 ◽  
Author(s):  
Domingos M. Cardoso ◽  
Dragoš Cvetković ◽  
Peter Rowlinson ◽  
Slobodan K. Simić
Keyword(s):  
Lower Bound ◽  
Bipartite Graph ◽  
Least Eigenvalue ◽  
Signless Laplacian ◽  
Sharp Lower Bound

scholarly journals Complexidade Parametrizada de Cliques e Conjuntos Independentes em Grafos Prismas Complementares

2018 ◽  
Author(s):  
Priscila Camargo ◽  
Alan D. A. Carneiro ◽  
Uéverton S. Santos
Keyword(s):  
Disjoint Union ◽  
Perfect Matching ◽  
Independent Set ◽  
Point Of View ◽  
Polynomial Kernel ◽  
Input Graph ◽  
Fixed Parameter Tractable ◽  
Fixed Parameter ◽  
Np Complete ◽  
Complementary Prism

The complementary prism GG¯ arises from the disjoint union of the graph G and its complement G¯ by adding the edges of a perfect matching joining pairs of corresponding vertices of G and G¯. The classical problems of graph theory, clique and independent set were proved NP-complete when the input graph is a complemantary prism. In this work, we study the complexity of both problems in complementary prisms graphs from the parameterized complexity point of view. First, we prove that these problems have a kernel and therefore are Fixed-Parameter Tractable (FPT). Then, we show that both problems do not admit polynomial kernel.

scholarly journals On the harmonic index of bicyclic conjugated molecular graphs

Filomat ◽  
2014 ◽  
Vol 28 (2) ◽  
pp. 421-428 ◽  
Author(s):  
Yan Zhu ◽  
Renying Chang
Keyword(s):  
Lower Bound ◽  
Perfect Matching ◽  
Harmonic Index ◽  
Molecular Graphs ◽  
Degree Of A Vertex ◽  
Bicyclic Graphs ◽  
Sharp Lower Bound ◽  
Given Size Of Matching

The harmonic index H(G) of a graph G is defined as the sum of weights 2/ d(u)+d(v) of all edges uv of G, where d(u) denotes the degree of a vertex u in G. In this paper, we first present a sharp lower bound on the harmonic index of bicyclic conjugated molecular graphs (bicyclic graphs with perfect matching). Also a sharp lower bound on the harmonic index of bicyclic graphs is given in terms of the order and given size of matching.

The least distance between extremums and minimal period of solutions to autonomous vector differential equations

2019 ◽  
Vol 485 (2) ◽  
pp. 142-144
Author(s):  
A. A. Zevin
Keyword(s):  
Lower Bound ◽  
Differential Equations ◽  
Periodic Solutions ◽  
Lipschitz Constant ◽  
Minimal Period ◽  
Arbitrary Vector ◽  
Vector Norm ◽  
Sharp Lower Bound

Solutions x(t) of the Lipschitz equation x = f(x) with an arbitrary vector norm are considered. It is proved that the sharp lower bound for the distances between successive extremums of xk(t) equals π/L where L is the Lipschitz constant. For non-constant periodic solutions, the lower bound for the periods is 2π/L. These estimates are achieved for norms that are invariant with respect to permutation of the indices.

On the Crossing Number of $K_{m,n}$

10.37236/1748 ◽  
2003 ◽  
Vol 10 (1) ◽  
Author(s):  
Nagi H. Nahas
Keyword(s):  
Lower Bound ◽  
Bipartite Graph ◽  
Complete Bipartite Graph ◽  
Crossing Number

The best lower bound known on the crossing number of the complete bipartite graph is : $$cr(K_{m,n}) \geq (1/5)(m)(m-1)\lfloor n/2 \rfloor \lfloor(n-1)/2\rfloor$$ In this paper we prove that: $$cr(K_{m,n}) \geq (1/5)m(m-1)\lfloor n/2 \rfloor \lfloor (n-1)/2 \rfloor + 9.9 \times 10^{-6} m^2n^2$$ for sufficiently large $m$ and $n$.

scholarly journals Spectral Gap of the Largest Eigenvalue of the Normalized Graph Laplacian

2021 ◽  
Author(s):  
Jürgen Jost ◽  
Raffaella Mulas ◽  
Florentin Münch
Keyword(s):  
Lower Bound ◽  
Bipartite Graph ◽  
Spectral Gap ◽  
Complete Bipartite Graph ◽  
Graph Laplacian ◽  
New Method ◽  
Single Edge ◽  
Largest Eigenvalue ◽  
Complement Graph ◽  
The Largest Eigenvalue

AbstractWe offer a new method for proving that the maxima eigenvalue of the normalized graph Laplacian of a graph with n vertices is at least $$\frac{n+1}{n-1}$$ n + 1 n - 1 provided the graph is not complete and that equality is attained if and only if the complement graph is a single edge or a complete bipartite graph with both parts of size $$\frac{n-1}{2}$$ n - 1 2 . With the same method, we also prove a new lower bound to the largest eigenvalue in terms of the minimum vertex degree, provided this is at most $$\frac{n-1}{2}$$ n - 1 2 .

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