Paths of Length Three are $K_{r+1}$-Turán-Good

The generalized Turán problem ex$(n,T,F)$ is to determine the maximal number of copies of a graph $T$ that can exist in an $F$-free graph on $n$ vertices. Recently, Gerbner and Palmer noted that the solution to the generalized Turán problem is often the original Turán graph. They gave the name "$F$-Turán-good" to graphs $T$ for which, for large enough $n$, the solution to the generalized Turán problem is realized by a Turán graph. They prove that the path graph on two edges, $P_2$, is $K_{r+1}$-Turán-good for all $r \ge 3$, but they conjecture that the same result should hold for all $P_\ell$. In this paper, using arguments based in flag algebras, we prove that the path on three edges, $P_3$, is also $K_{r+1}$-Turán-good for all $r \ge 3$.

Optimal Linear Arrangement of Turán Graphs

Graph embedding in parallel processing techniques has acquired considerable attention and hence raised as an efficient approach for reducing overhead data into low-dimensional space. Optimal layout and congestion are powerful parameters to examine the capability of embedding. In this study, Modified Congestion and  -Partition lemmas are utilized to obtain the optimal layout of Turán graph into path and windmill graphs.

Extremal Matching Energy and the Largest Matching Root of Complete Multipartite Graphs

The matching energy ME(G) of a graph G was introduced by Gutman and Wagner, which is defined as the sum of the absolute values of the roots of the matching polynomial m(G,x). The largest matching root λ1(G) is the largest root of the matching polynomial m(G,x). Let Kn1,n2,…,nr denote the complete r-partite graph with order n=n1+n2+…+nr, where r>1. In this paper, we prove that, for the given values n and r, both the matching energy ME(G) and the largest matching root λ1(G) of complete r-partite graphs are minimal for complete split graph CS(n,r-1) and are maximal for Turán graph T(n,r).

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

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].

Extremal Graph for Intersecting Odd Cycles

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.

Extremal Numbers for Odd Cycles

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.

An Extension of Turán's Theorem, Uniqueness and Stability

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.

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

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 $r.$

Spectral Saturation: Inverting the Spectral Turán Theorem

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.