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Register a new userRemarks on the spectral radius of $K_{r+1}$-saturated graphs
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V. Nikiforov
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Write $rholeft( Gright) $ for the spectral radius of a graph $G$ and $S_{n,r}$ for the join $K_{r}veeoverline{K}_{n-r}.$ Let $n>rgeq2$ and $G$ be a $K_{r+1}$-saturated graph of order $n.$ Recently Kim, Kim, Kostochka, and O determined exactly the minimum value of $rholeft( Gright) $ for $r=2$, and found an asymptotically tight bound on $rholeft( Gright) $ for $rgeq3.$ They also conjectured that [ rholeft( Gright) >rholeft( S_{n,r-1}right) , ] unless $G=S_{n,r-1}.$ In this note their conjecture is proved.
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For a graph $H$, a graph $G$ is $H$-saturated if $G$ does not contain $H$ as a subgraph but for any $e in E(overline{G})$, $G+e$ contains $H$. In this note, we prove a sharp lower bound for the number of paths and walks on length $2$ in $n$-vertex $K_{r+1}$-saturated graphs. We then use this bound to give a lower bound on the spectral radii of such graphs which is asymptotically tight for each fixed $r$ and $ntoinfty$.
The ErdH{o}s-Simonovits stability theorem states that for all epsilon >0 there exists alpha >0 such that if G is a K_{r+1}-free graph on n vertices with e(G) > ex(n,K_{r+1}) - alpha n^2, then one can remove epsilon n^2 edges from G to obtain an r-partite graph. Furedi gave a short proof that one can choose alpha=epsilon. We give a bound for the relationship of alpha and varepsilon which is asymptotically sharp as epsilon to 0.
Let $tgeq3$ and $G$ be a graph of order $n,$ with no $K_{2,t}$ minor. If $n>400t^{6}$, then the spectral radius $muleft( Gright) $ satisfies [ muleft( Gright) leqfrac{t-1}{2}+sqrt{n+frac{t^{2}-2t-3}{4}}, ] with equality if and only if $nequiv1$ $(operatorname{mod}$ $t)$ and $G=K_{1}veeleftlfloor n/trightrfloor K_{t}$. For $t=3$ the maximum $muleft( Gright) $ is found exactly for any $n>40000$.
This paper considers an edge minimization problem in saturated bipartite graphs. An $n$ by $n$ bipartite graph $G$ is $H$-saturated if $G$ does not contain a subgraph isomorphic to $H$ but adding any missing edge to $G$ creates a copy of $H$. More than half a century ago, Wessel and Bollobas independently solved the problem of minimizing the number of edges in $K_{(s,t)}$-saturated graphs, where $K_{(s,t)}$ is the `ordered complete bipartite graph with $s$ vertices from the first color class and $t$ from the second. However, the very natural `unordered analogue of this problem was considered only half a decade ago by Moshkovitz and Shapira. When $s=t$, it can be easily checked that the unordered variant is exactly the same as the ordered case. Later, Gan, Korandi, and Sudakov gave an asymptotically tight bound on the minimum number of edges in $K_{s,t}$-saturated $n$ by $n$ bipartite graphs, which is only smaller than the conjecture of Moshkovitz and Shapira by an additive constant. In this paper, we confirm their conjecture for $s=t-1$ with the classification of the extremal graphs. We also improve the estimates of Gan, Korandi, and Sudakov for general $s$ and $t$, and for all sufficiently large $n$.
Let $K_{m}-H$ be the graph obtained from $K_{m}$ by removing the edges set $E(H)$ of the graph $H$ ($H$ is a subgraph of $K_{m}$). We use the symbol $Z_4$ to denote $K_4-P_2.$ A sequence $S$ is potentially $K_{m}-H$-graphical if it has a realization containing a $K_{m}-H$ as a subgraph. Let $sigma(K_{m}-H, n)$ denote the smallest degree sum such that every $n$-term graphical sequence $S$ with $sigma(S)geq sigma(K_{m}-H, n)$ is potentially $K_{m}-H$-graphical. In this paper, we determine the values of $sigma (K_{r+1}-U, n)$ for $ngeq 5r+18, r+1 geq k geq 7,$ $j geq 6$ where $U$ is a graph on $k$ vertices and $j$ edges which contains a graph $K_3 bigcup P_3$ but not contains a cycle on 4 vertices and not contains $Z_4$. There are a number of graphs on $k$ vertices and $j$ edges which contains a graph $(K_{3} bigcup P_{3})$ but not contains a cycle on 4 vertices and not contains $Z_4$. (for example, $C_3bigcup C_{i_1} bigcup C_{i_2} bigcup >... bigcup C_{i_p}$ $(i_j eq 4, j=2,3,..., p, i_1 geq 5)$, $C_3bigcup P_{i_1} bigcup P_{i_2} bigcup ... bigcup P_{i_p}$ $(i_1 geq 3)$, $C_3bigcup P_{i_1} bigcup C_{i_2} bigcup >... bigcup C_{i_p}$ $(i_j eq 4, j=2,3,..., p, i_1 geq 3)$, etc)
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