Abstract
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.
Highlights
In this paper, we consider only simple and undirected graphs
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 Fk
We prove the theorem iteratively, giving successively better lower bounds on both e(G) and the eigenvector entries of all of the other vertices, until we can show that e(G) = ex(n, Fk)
Summary
We consider only simple and undirected graphs. Let G be a simple connected graph with vertex set V (G) = {v1, . . . , vn} and edge set E(G) = {e1, . . . , em}. Brualdi and Solheid [5] proposed the following problem: Given a set of graphs, try to find a tight upper bound for the spectral radius in this set and characterize all extremal graphs. Nikiforov obtained spectral strengthenings of Turan’s theorem [24] and the Kovari-Sos-Turan theorem [22] when the forbidden graphs are complete or complete bipartite respectively. A graph on 2k + 1 vertices consisting of k triangles which intersect in exactly one common vertex is called a k−friendship graph ( known as a k-fan) and denoted by Fk. In [10], the following result is proved. We at last point out that, during our proof, we use the triangle removal lemma, so it is difficult to present exactly how large we need our n to be
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