scholarly journals Extremal Graph for Intersecting Odd Cycles

10.37236/5851 ◽  
2016 ◽  
Vol 23 (2) ◽  
Author(s):  
Xinmin Hou ◽  
Yu Qiu ◽  
Boyuan Liu
Keyword(s):  
Complete Graph ◽  
Extremal Graph ◽  
Extremal Graphs ◽  
Turán Theorem ◽  
Odd Cycles ◽  
Turán Graph

An extremal graph for a graph $H$ on $n$ vertices is a graph on $n$ vertices with maximum number of edges that does not contain $H$ as a subgraph. Let $T_{n,r}$ be the Turán graph, which is the complete $r$-partite graph on $n$ vertices with part sizes that differ by at most one. The well-known Turán Theorem states that $T_{n,r}$ is the only extremal graph for complete graph $K_{r+1}$. Erdős et al. (1995) determined the extremal graphs for intersecting triangles and Chen et al. (2003) determined the maximum number of edges of the extremal graphs for intersecting cliques. In this paper, we determine the extremal graphs for intersecting odd cycles.

scholarly journals Extremal Numbers for Odd Cycles

2014 ◽  
Vol 24 (4) ◽  
pp. 641-645 ◽  
Author(s):  
ZOLTAN FÜREDI ◽  
DAVID S. GUNDERSON
Keyword(s):  
Extremal Graph ◽  
Classical Theorem ◽  
Extremal Graphs ◽  
Complete Determination ◽  
Odd Cycles ◽  
Turán Graph

We describe theC2k+1-free graphs onnvertices with maximum number of edges. The extremal graphs are unique forn∉ {3k− 1, 3k, 4k− 2, 4k− 1}. The value ofex(n,C2k+1) can be read out from the works of Bondy [3], Woodall [14], and Bollobás [1], but here we give a new streamlined proof. The complete determination of the extremal graphs is also new.We obtain that the bound forn0(C2k+1) is 4kin the classical theorem of Simonovits, from which the unique extremal graph is the bipartite Turán graph.

scholarly journals An Extension of Turán's Theorem, Uniqueness and Stability

10.37236/4194 ◽  
2014 ◽  
Vol 21 (4) ◽  
Author(s):  
Peter Allen ◽  
Julia Böttcher ◽  
Jan Hladký ◽  
Diana Piguet
Keyword(s):  
Extremal Graph ◽  
Extremal Graphs ◽  
Fixed Set ◽  
Vertex Graph ◽  
Turan's Theorem ◽  
Turán’S Theorem ◽  
Stability Results ◽  
Turán Graph

We determine the maximum number of edges of an $n$-vertex graph $G$ with the property that none of its $r$-cliques intersects a fixed set $M\subset V(G)$.  For $(r-1)|M|\ge n$, the $(r-1)$-partite Turán graph turns out to be the unique extremal graph. For $(r-1)|M|<n$, there is a whole family of extremal graphs, which we describe explicitly. In addition we provide corresponding stability results.

scholarly journals Spectral Saturation: Inverting the Spectral Turán Theorem

10.37236/122 ◽  
2009 ◽  
Vol 16 (1) ◽  
Author(s):  
Vladimir Nikiforov
Keyword(s):  
Complete Graph ◽  
Largest Eigenvalue ◽  
Turán Theorem ◽  
Stability Results ◽  
Turán Graph ◽  
The Largest Eigenvalue

Let $\mu\left( G\right) $ be the largest eigenvalue of a graph $G$ and $T_{r}\left( n\right) $ be the $r$-partite Turán graph of order $n.$We prove that if $G$ is a graph of order $n$ with $\mu\left( G\right)>\mu\left( T_{r}\left( n\right) \right)$, then $G$ contains various large supergraphs of the complete graph of order $r+1,$ e.g., the complete $r$-partite graph with all parts of size $\log n$ with an edge added to the first part.We also give corresponding stability results.

scholarly journals On the Generalized Distance Energy of Graphs

Mathematics ◽  
2019 ◽  
Vol 8 (1) ◽  
pp. 17 ◽  
Author(s):  
Abdollah Alhevaz ◽  
Maryam Baghipur ◽  
Hilal A. Ganie ◽  
Yilun Shang
Keyword(s):  
Lower Bounds ◽  
Complete Graph ◽  
Connected Graph ◽  
Distance Matrix ◽  
Wiener Index ◽  
Diagonal Matrix ◽  
Upper And Lower Bounds ◽  
Star Graph ◽  
Extremal Graphs ◽  
Generalized Distance

The generalized distance matrix D α ( G ) of a connected graph G is defined as D α ( G ) = α T r ( G ) + ( 1 − α ) D ( G ) , where 0 ≤ α ≤ 1 , D ( G ) is the distance matrix and T r ( G ) is the diagonal matrix of the node transmissions. In this paper, we extend the concept of energy to the generalized distance matrix and define the generalized distance energy E D α ( G ) . Some new upper and lower bounds for the generalized distance energy E D α ( G ) of G are established based on parameters including the Wiener index W ( G ) and the transmission degrees. Extremal graphs attaining these bounds are identified. It is found that the complete graph has the minimum generalized distance energy among all connected graphs, while the minimum is attained by the star graph among trees of order n.

scholarly journals On the Turán Number of Forests

10.37236/3142 ◽  
2013 ◽  
Vol 20 (2) ◽  
Author(s):  
Hong Liu ◽  
Bernard Lidicky ◽  
Cory Palmer
Keyword(s):  
Arbitrary Order ◽  
Extremal Graph ◽  
Extremal Graphs ◽  
Turán Number

The Turán number of a graph $H$, $\mathrm{ex}(n,H)$, is the maximum number of edges in a graph on $n$ vertices which does not have $H$ as a subgraph. We determine the Turán number and find the unique extremal graph for forests consisting of paths when $n$ is sufficiently large. This generalizes a result of Bushaw and Kettle [Combinatorics, Probability and Computing 20:837--853, 2011]. We also determine the Turán number and extremal graphs for forests consisting of stars of arbitrary order.

scholarly journals Invariants of the symbolic powers of edge ideals

2019 ◽  
Vol 19 (10) ◽  
pp. 2050184
Author(s):  
Bidwan Chakraborty ◽  
Mousumi Mandal
Keyword(s):  
Complete Graph ◽  
Explicit Description ◽  
Edge Ideal ◽  
Edge Ideals ◽  
Single Vertex ◽  
Symbolic Powers ◽  
Odd Cycles ◽  
Waldschmidt Constant

Let [Formula: see text] be a graph and [Formula: see text] be its edge ideal. When [Formula: see text] is the clique sum of two different length odd cycles joined at single vertex then we give an explicit description of the symbolic powers of [Formula: see text] and compute the Waldschmidt constant. When [Formula: see text] is complete graph then we describe the generators of the symbolic powers of [Formula: see text] and compute the Waldschmidt constant and the resurgence of [Formula: see text]. Moreover for complete graph we prove that the Castelnuovo–Mumford regularity of the symbolic powers and ordinary powers of the edge ideal coincide.

scholarly journals Extremal graph problems with symmetrical extremal graphs. Additional chromatic conditions

1974 ◽  
Vol 7 (3-4) ◽  
pp. 349-376 ◽  
Author(s):  
M. Simonovits
Keyword(s):  
Extremal Graph ◽  
Extremal Graphs ◽  
Graph Problems

scholarly journals On some interconnections between combinatorial optimization and extremal graph theory

2004 ◽  
Vol 14 (2) ◽  
pp. 147-154 ◽  
Author(s):  
Dragos Cvetkovic ◽  
Pierre Hansen ◽  
Vera Kovacevic-Vujcic
Keyword(s):  
Graph Theory ◽  
Combinatorial Optimization ◽  
Efficient Algorithms ◽  
Extremal Graph ◽  
Extremal Graph Theory ◽  
Extremal Graphs ◽  
Graph Invariants ◽  
Finite Set

The uniting feature of combinatorial optimization and extremal graph theory is that in both areas one should find extrema of a function defined in most cases on a finite set. While in combinatorial optimization the point is in developing efficient algorithms and heuristics for solving specified types of problems, the extremal graph theory deals with finding bounds for various graph invariants under some constraints and with constructing extremal graphs. We analyze by examples some interconnections and interactions of the two theories and propose some conclusions.

Extremal graph on normalized Laplacian spectral radius and energy

2015 ◽  
Vol 29 ◽  
pp. 237-253 ◽  
Author(s):  
Kinkar Das ◽  
SHAOWEI SUN
Keyword(s):  
Lower Bound ◽  
Spectral Radius ◽  
Complete Graph ◽  
Connected Graph ◽  
Simple Graph ◽  
Extremal Graphs ◽  
Laplacian Spectral Radius ◽  
Laplacian Eigenvalues ◽  
Laplacian Energy ◽  
Isolated Vertex

Let $G=(V,\,E)$ be a simple graph of order $n$ and the normalized Laplacian eigenvalues $\rho_1\geq \rho_2\geq \cdots\geq\rho_{n-1}\geq \rho_n=0$. The normalized Laplacian energy (or Randi\'c energy) of $G$ without any isolated vertex is defined as $$RE(G)=\sum_{i=1}^{n}|\rho_i-1|.$$ In this paper, a lower bound on $\rho_1$ of connected graph $G$ ($G$ is not isomorphic to complete graph) is given and the extremal graphs (that is, the second minimal normalized Laplacian spectral radius of connected graphs) are characterized. Moreover, Nordhaus-Gaddum type results for $\rho_1$ are obtained. Recently, Gutman et al.~gave a conjecture on Randi\'c energy of connected graph [I. Gutman, B. Furtula, \c{S}. B. Bozkurt, On Randi\'c energy, Linear Algebra Appl. 442 (2014) 50--57]. Here this conjecture for starlike trees is proven.

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