New Bounds for the α-Indices of Graphs - Eber Lenes; Exequiel Mallea-Zepeda; Jonnathan Rodríguez (Mathematics, 2020)
Let G be a graph, for any real <semantics>0≤α≤1</semantics>, Nikiforov defines the matrix <semantics>Aα(G)</semantics> as <semantics>Aα(G)=αD(G)+(1−α)A(G),</semantics> where <semantics>A(G)</semantics> and <semantics>D(G)</semantics> are the adjacency matrix and diagonal matrix of degrees of the vertices of G. This paper presents some extremal results about the spectral radius <semantics>ρα(G)</semantics> of the matrix <semantics>Aα(G)</semantics>. In particular, we give a lower bound on the spectral radius <semantics>ρα(G)</semantics> in terms of order and independence number. In addition, we obtain an upper bound for the spectral radius <semantics>ρα(G)</semantics> in terms of order and minimal degree. Furthermore, for <semantics>n>l>0</semantics> and <semantics>1≤p≤⌊n−l2⌋</semantics>, let <semantics>Gp≅Kl∨(Kp∪Kn−p−l)</semantics> be the graph obtained from the graphs <semantics>Kl</semantics> and <semantics>Kp∪Kn−p−l</semantics> and edges connecting each vertex of <semantics>Kl</semantics> with every vertex of <semantics>Kp∪Kn−p−l.</semantics> We prove that <semantics>ρα(Gp+1)<ρα(Gp)</semantics> for <semantics>1≤p≤⌊n−l2⌋−1</semantics>.