On the spectral radius of graphs

Bounds for the Laplacian spectral radius of graphs

“…As an application of Theorem 4.17 together with the Cauchy-Schwarz inequality, the following bound is obtained in [54,Corollary 10], also see [20,Theorem 3.2]. …”

Graphs with Equal Irregularity Indices

“…A bipartite graph G is semiregular if every edge of G joins a vertex of degree δ to a vertex of degree ∆. The 2-degree of a vertex u, denoted by d 2 (u) is the sum of degrees of the vertices adjacent to u [20]. The average-degree of u is d 2 (u)/d(u) and it is denoted by p(u).…”

“…A graph G is called pseudo-regular (or harmonic) if every vertex of G has equal average-degree. A bipartite graph is called pseudo-semiregular if each vertex in the same part of a bipartition has the same average-degree [20]. It follows that semiregular graphs form a subset of pseudo-semiregular graphs.…”

On Some Properties of Pseudo-Semiregular Graphs

“…A connected graph G is said to be harmonic (pseudo-regular) [2,3,4] if there exists a positive constant p(G) such that each vertex u of G has the same average neighbor degree number [2,3] identical with p(G). The spectral radius of a harmonic graph G is equal to p(G).…”

“…The spectral radius of a harmonic graph G is equal to p(G). It is obvious [2] that any connected R-regular graph G R is a harmonic graph with p(G R )= ρ(G R )=R. A connected bipartite graph is called pseudo-semiregular [2,5] if each vertex in the same part of bipartition has the same average degree.…”