In this paper, we examine the structure of strongly regular graphs with parameters $\lambda = 1$ and $\mu = 2$. In particular, we provide a complete classification of induced subgraphs of order seven and determine their relative frequencies. These findings contribute to a finer understanding of the local structure of such graphs and may be useful in related combinatorial and algebraic investigations.
For graphs $H$ and $F$, let $\operatorname{ex}(n,H,F)$ denote the maximum number of copies of $H$ in an $F$-free graph of order $n$. Motivated by the Erd\H{o}s-S\'{o}s conjecture, Gerbner and Palmer and, independently, Zhao and Peng conjectured that for every tree $T$ of order $k$ and every $3\le r\le k-1$, $$\operator...
Given two weighted directed graphs of order \(n\), we study how much their overlap can deviate from its random average under relabelling, and how concentrated the distribution of their intersection is when they are placed at random. Bollob\'as and Scott studied these problems on weighted $k$-uniform hypergraphs and rai...
We show that if an n-vertex 5-connected graph has at least 4n-10 edges, then for any choice of five of its vertices, we can contract disjoint connected subgraphs containing these vertices to obtain $K_5$ as a minor. The bound on the number of edges is the best possible.
We prove that, for every integer $k\ge 2$, there exists a constant $c_k>0$ such that every graph on $n\ge R(k,k)$ vertices with maximum degree $\Delta$ contains an induced subgraph on at least $n-c_k\sqrt{\Delta}$ vertices whose maximum degree is attained by at least $k$ vertices. This confirms a conjecture of Caro and...
How many disjoint copies of some graph $H$ can appear in an induced subgraph of a strongly regular graph? We give upper and lower bounds that depend on the number of vertices in their boundary. This generalizes a result of Lou-Murin which bounds the independence number of a (hitherto still elusive) strongly regular gra...
For a finite simple graph $G$, let $\nu(G)$ be the maximum number of pairwise edge disjoint triangles and $\tau(G)$ the minimum number of edges meeting every triangle. Tuza conjectured in 1981 that every such $G$ satisfies $\tau(G) \le 2 \nu(G).$ We prove the conjecture for the class of graphs with $\Delta\le10.$
Amrit Kandasamy· 0 citations
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