Upper bounds on the spectral radius of book-free and/or \(K_{2,l}\)-free graphs
In this paper, the authors obtain upper bounds on the spectral radius of book-free and/or \(K_{2,l}\)-free graphs. Let \(G=(V,E)\) be a graph of order \(n\). The spectral radius \(\rho(G)\) of \(G\) is the largest eigenvalue of its adjacency matrix. For \(v \in V\), the degree of \(v\) is \(d(v)=| N(v)| \), where \(N(v)\) is the set of vertices adjacent to \(v\). The maximum degree of \(G\) is \(\Delta(G)=\max\{d(v) : v \in V \}\). A \(k\)-regular graph, that is a graph whose vertices have the same degree \(k\), is called strongly regular with parameters \((k,a,c)\) whenever each pair of adjacent vertices have \(a \geq 0\) common neighbors, and each pair of nonadjacent vertices have \(c \geq 1\) common neighbors. The main result of this paper establishs that if \(0\leq k \leq l \leq \Delta < n\) and \(G\) is a connected \(\{B_{k+1},K_{2,l+1}\}\)-free graph of order \(n\) with maximum degree \(\Delta\), then \[ \rho(G) \leq \left[k-l+\sqrt{(k-l)^2+4 \Delta +4l(n-1)} \right]/2, \] where \(B_{k+1}\) denotes a book with \(k+1\) pages consisting of \(k+1\) triangles sharing one edge. The equality, in the above expression, is satisfied if and only if \(G\) is a strongly regular graph with parameters \((\Delta, k, l)\).
scientific article; zbMATH DE number 5083621