9 problems

Let $F_k$ be the friendship graph consisting of $k$ triangles sharing a common vertex. Let $G$ be an $F_k$-free graph with $m$ edges, and let $\rho(G)$ denote its spectral radius.…

Let $F_k$ be the friendship graph with $k$ triangles sharing a common vertex. For $k\ge1$ and $n\ge50k^2$, the source records … Erdős–Füredi–Gould–Gunderson's conjecture. The same…

Let $F_n$ be the friendship graph formed by joining $n$ copies of $C_3$ at a central vertex. Let $m,n\in\mathbb{N}$ with $m\geq 1$. The known obstruction is that $F_n$ is not…

Let $F_n$ be the friendship graph formed by joining $n$ copies of $C_3$ at a central vertex. For $m,n\in\mathbb{N}$ with $m\geq 1$, suppose that $F_n$ is obstructed from being…

Let $F_n$ be a friendship graph, and let $G$ be a graph cospectral with $F_n$. Wang–Zhao–Huang conjecture. The minimum degree of $G$ is $2$. The source notes that the proofs in the…

Let $F_k$ be the friendship graph consisting of $k$ edge-disjoint triangles sharing a common vertex. A graph is determined by its adjacency spectrum if every graph with the same ad…

Friendship graph spectral-determination conjecture. The friendship graph is $DS$; equivalently, if a graph $H$ has the same adjacency spectrum as $F_n$, then $H\cong F_n$.

Let $F_k$ be the friendship graph consisting of $k$ edge-disjoint triangles meeting in one vertex, where $k\geq 2$, and let $A_k$ be its adjacency matrix. Its spectrum is … Wang–Be…

Let $F_n$ be the friendship graph consisting of $n$ triangles sharing a common vertex. A graph is $DS$ with respect to the adjacency matrix if it is determined up to isomorphism by…