The number of maximal cliques and spectral radius of graphs with certain forbidden subgraphs

2018 ◽  
Vol 10 (06) ◽  
pp. 1850071
Author(s):  
Ya-Lei Jin ◽  
Xiao-Dong Zhang
Keyword(s):  
Spectral Radius ◽  
Forbidden Subgraphs ◽  
Unique Graph ◽  
Turán Theorem ◽  
Turán Graph

Turán theorem states that the Turán graph [Formula: see text] is the unique graph which has the maximum edge number among the [Formula: see text]-free graphs of order [Formula: see text]. In this paper, we prove that [Formula: see text] has both the maximum number of maximal cliques and the maximum spectral radius among all graphs of order [Formula: see text] with [Formula: see text] and [Formula: see text], where [Formula: see text] stands for the maximum number of disjoint [Formula: see text]cliques of [Formula: see text].

scholarly journals On the spectral radius of cactuses with perfect matchings

2011 ◽  
Vol 5 (1) ◽  
pp. 14-21 ◽  
Author(s):  
Ziwen Huang ◽  
Hanyuan Deng ◽  
Slobodan Simic
Keyword(s):  
Spectral Radius ◽  
Perfect Matchings ◽  
Unique Graph

Let C(2m, k) be the set of all cactuses on 2m vertices, k cycles, and with perfect matchings. In this paper, we identify in C(2m, k) the unique graph with the largest spectral radius.

SPECTRAL RADII OF UNICYCLIC GRAPHS WITH FIXED NUMBER OF CUT VERTICES

2011 ◽  
Vol 03 (02) ◽  
pp. 139-145 ◽  
Author(s):  
YI-ZHENG FAN ◽  
JING-MEI ZHANG ◽  
YI WANG
Keyword(s):  
Spectral Radius ◽  
Fixed Number ◽  
Unicyclic Graphs ◽  
Unique Graph

Let [Formula: see text] be the set of unicyclic graphs with n vertices and k cut vertices. In this paper, we determine the unique graph with the maximal spectral radius among all graphs in [Formula: see text] for 1 ≤ k ≤ n - 3.

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 Degree Powers in Graphs with Forbidden Subgraphs

10.37236/1795 ◽  
2004 ◽  
Vol 11 (1) ◽  
Author(s):  
Béla Bollobás ◽  
Vladimir Nikiforov
Keyword(s):  
Simple Graph ◽  
Forbidden Subgraphs ◽  
Turán Graph

For every real $p>0$ and simple graph $G,$ set $$ f\left( p,G\right) =\sum_{u\in V\left( G\right) }d^{p}\left( u\right) , $$ and let $\phi\left( r,p,n\right) $ be the maximum of $f\left( p,G\right) $ taken over all $K_{r+1}$-free graphs $G$ of order $n.$ We prove that, if $0 < p < r,$ then$$ \phi\left( r,p,n\right) =f\left( p,T_{r}\left( n\right) \right) , $$ where $T_{r}\left( n\right) $ is the $r$-partite Turan graph of order $n$. For every $p\geq r+\left\lceil \sqrt{2r}\right\rceil $ and $n$ large, we show that$$ \phi\left( p,n,r\right) >\left( 1+\varepsilon\right) f\left( p,T_{r}\left( n\right) \right) $$ for some $\varepsilon=\varepsilon\left( r\right) >0.$ Our results settle two conjectures of Caro and Yuster.

scholarly journals Extremal Problems for the $p$-Spectral Radius of Graphs

10.37236/4113 ◽  
2014 ◽  
Vol 21 (3) ◽  
Author(s):  
Liying Kang ◽  
Vladimir Nikiforov
Keyword(s):  
Real Number ◽  
Spectral Radius ◽  
Extremal Problems ◽  
Free Graph ◽  
Remarkable Feature ◽  
Graph Parameters ◽  
Turán Graph

The $p$-spectral radius of a graph $G\ $of order $n$ is defined for any real number $p\geq1$ as$$\lambda^{(p)}(G) =\max\{ 2\sum_{\{i,j\}\in E(G)} x_ix_j:x_1,\ldots,x_n\in\mathbb{R}\text{ and }\vert x_{1}\vert ^{p}+\cdots+\vert x_n\vert^{p}=1\} .$$The most remarkable feature of $\lambda^{(p)}$ is that it seamlessly joins several other graph parameters, e.g., $\lambda^{(1)}$ is the Lagrangian, $\lambda^{(2)  }$ is the spectral radius and $\lambda^{(\infty)  }/2$ is the number of edges. This paper presents solutions to some extremal problems about $\lambda^{(p)}$, which are common generalizations of corresponding edge and spectral extremal problems.Let $T_{r}\left(  n\right)  $ be the $r$-partite Turán graph of order $n$. Two of the main results in the paper are:(I) Let $r\geq2$ and $p>1.$ If $G$ is a $K_{r+1}$-free graph of order $n$, then$$\lambda^{(p)}(G)  <\lambda^{(p)}(T_{r}(n)),$$ unless $G=T_{r}(n)$.(II) Let $r\geq2$ and $p>1.$ If $G\ $is a graph of order $n,$ with$$\lambda^{(p)}(G)>\lambda^{(p)}(  T_{r}(n))  ,$$then $G$ has an edge contained in at least $cn^{r-1}$ cliques of order $r+1$, where $c$ is a positive number depending only on $p$ and $r.$

Induced Turán Numbers

2017 ◽  
Vol 27 (2) ◽  
pp. 274-288 ◽  
Author(s):  
PO-SHEN LOH ◽  
MICHAEL TAIT ◽  
CRAIG TIMMONS ◽  
RODRIGO M. ZHOU
Keyword(s):  
Upper Bound ◽  
Independent Interest ◽  
Forbidden Subgraphs ◽  
Free Graph ◽  
Induced Subgraph ◽  
Vertex Graph ◽  
Recent Theorem ◽  
Fixed Order ◽  
Turán Theorem ◽  
Asymptotic Upper Bound

The classical Kővári–Sós–Turán theorem states that ifGis ann-vertex graph with no copy ofKs,tas a subgraph, then the number of edges inGis at mostO(n2−1/s). We prove that if one forbidsKs,tas aninducedsubgraph, and also forbidsanyfixed graphHas a (not necessarily induced) subgraph, the same asymptotic upper bound still holds, with different constant factors. This introduces a non-trivial angle from which to generalize Turán theory to induced forbidden subgraphs, which this paper explores. Along the way, we derive a non-trivial upper bound on the number of cliques of fixed order in aKr-free graph with no induced copy ofKs,t. This result is an induced analogue of a recent theorem of Alon and Shikhelman and is of independent interest.

A NOTE ON THE DISTANCE SPECTRAL RADIUS OF SOME GRAPHS

2014 ◽  
Vol 06 (01) ◽  
pp. 1450015 ◽  
Author(s):  
MILAN NATH ◽  
SOMNATH PAUL
Keyword(s):  
Spectral Radius ◽  
Minimum Degree ◽  
Clique Number ◽  
Minimal Distance ◽  
Vertex Connectivity ◽  
Unique Graph

We characterize graphs with minimal distance spectral radius in two classes of graphs: with vertex connectivity k and minimum degree at least k, and with given number of blocks. Moreover, we determine the unique graph that maximizes the distance spectral radius among all graphs with given clique number.

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 The Signless Laplacian Spectral Radius of Unicyclic and Bicyclic Graphs with a Given Girth

10.37236/670 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Ke Li ◽  
Ligong Wang ◽  
Guopeng Zhao
Keyword(s):  
Spectral Radius ◽  
Upper Bound ◽  
Extremal Graph ◽  
Laplacian Spectral Radius ◽  
Unicyclic Graphs ◽  
Signless Laplacian Spectral Radius ◽  
Disjoint Cycles ◽  
Signless Laplacian ◽  
Unique Graph ◽  
Bicyclic Graphs

Let $\mathcal{U}(n,g)$ and $\mathcal{B}(n,g)$ be the set of unicyclic graphs and bicyclic graphs on $n$ vertices with girth $g$, respectively. Let $\mathcal{B}_{1}(n,g)$ be the subclass of $\mathcal{B}(n,g)$ consisting of all bicyclic graphs with two edge-disjoint cycles and $\mathcal{B}_{2}(n,g)=\mathcal{B}(n,g)\backslash\mathcal{B}_{1}(n,g)$. This paper determines the unique graph with the maximal signless Laplacian spectral radius among all graphs in $\mathcal{U}(n,g)$ and $\mathcal{B}(n,g)$, respectively. Furthermore, an upper bound of the signless Laplacian spectral radius and the extremal graph for $\mathcal{B}(n,g)$ are also given.

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