Graphic sequences with a realization containing intersecting cliques

Let r ≥ 1, k ≥ 2 and F_{m1 , ?mk ;r}F_{m_1 , \ldots m_k ;r} denote the most general definition of a friendship graph, that is, the graph of K_{r + m1} , ?,K_{r + mk} K_{r + m_1 } , \ldots ,K_{r + m_k } meeting in a common r set, where K_{r + m1} K_{r + m_1 } is the complete graph on r + m _i vertices. Clearly, | F_{m1 , ?mk ;r} | = m₁ + ?+ m_k + r\left| {F_{m_1 , \ldots m_k ;r} } \right| = m_1 + \cdots + m_k + r. Let s( F_{m1 , ?mk ;r} ,n )\sigma \left( {F_{m_1 , \ldots m_k ;r} ,n} \right) be the smallest even integer such that every n-term graphic sequence π = (d ₁, d ₂, ..., d _n ) with term sum s( p) = d₁ + d₂ + ?+ d_n \geqslant s( F_{m1 , ?mk ;r} ,n )\sigma \left( \pi \right) = d_1 + d_2 + \cdots + d_n \geqslant \sigma \left( {F_{m_1 , \ldots m_k ;r} ,n} \right), has a realization G containing F_{m1 , ?mk ;r}F_{m_1 , \ldots m_k ;r} as a subgraph. In this paper, we determine s( F_{m1 , ?mk ;r} ,n )\sigma \left( {F_{m_1 , \ldots m_k ;r} ,n} \right) for n sufficiently large.