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null (Ed.)A graph on $2k+1$ vertices consisting of $$k$$ triangles which intersect in exactly one common vertex is called a $k-$friendship graph and denoted by $$F_k$$. This paper determines the graphs of order $$n$$ that have the maximum (adjacency) spectral radius among all graphs containing no $$F_k$$, for $$n$$ sufficiently large.more » « less
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Let $$G$$ be a finite group acting transitively on $$[n]=\{1,2,\ldots,n\}$$, and let $$\Gamma=\mathrm{Cay}(G,T)$$ be a Cayley graph of $$G$$. The graph $$\Gamma$$ is called normal if $$T$$ is closed under conjugation. In this paper, we obtain an upper bound for the second (largest) eigenvalue of the adjacency matrix of the graph $$\Gamma$$ in terms of the second eigenvalues of certain subgraphs of $$\Gamma$$. Using this result, we develop a recursive method to determine the second eigenvalues of certain Cayley graphs of $$S_n$$, and we determine the second eigenvalues of a majority of the connected normal Cayley graphs (and some of their subgraphs) of $$S_n$$ with $$\max_{\tau\in T}|\mathrm{supp}(\tau)|\leqslant 5$$, where $$\mathrm{supp}(\tau)$$ is the set of points in $[n]$ non-fixed by $$\tau$$.more » « less
