Neighbourhood second Zagreb matrix of graphs: spectral radius bounds and extremal graphs

Let G be a connected graph on at least two vertices, and let $δ(i)$ be the sum of the degrees of the neighbours of a vertex i. The neighbourhood second Zagreb matrix (G) has $(i,j)$-entry $δ(i)δ(j)$ if i~ j and $0$ otherwise; its spectral radius is denoted by (G). We prove that (G)≥ρ(G)⁵, where $ρ(G)$ is the adjacency spectral radius, with equality if and only if G is regular or bipartite semiregular, and we use this to show that the same graphs are exactly the extremal graphs of a Berman--Zhang--type upper bound. We sharpen the estimates ²ρ(G)≤(G)≤²ρ(G) by means of the extreme edge weights of (G), give sharp upper bounds in terms of the maximum degree , the maximum neighbourhood degree sum , the size m and the clique number $ω(G)$, and determine the extremal graphs for bipartite graphs and trees. For every connected unicyclic graph G of order n≥3 we prove 32=(Cₙ)≤(G)≤( n), where n is the triangle with $n-3$ pendant edges at one vertex, with equality only for Cₙ and only for n, respectively. Finally, we give bounds in terms of the first Zagreb index and for the neighbourhood second Zagreb energy; for lower energy bounds, the extremal graphs are described completely.