Let Gσ be a graph obtained by attaching a self-loop, or just a loop, for short, at each of σ chosen vertices of a given graph G. Gutman et al. have recently introduced the concept of the energy of graphs with self-loops, and conjectured that the energy E(G) of a graph G of order n is always strictly less than the energy E(Gσ) of a corresponding graph Gσ, for 1 ≤ σ ≤ n − 1. In this paper, a simple set of graphs which disproves this conjecture is exposed, together with some remarks regarding the standard deviations of the (adjacency) eigenvalues of G and Gσ, respectively.

The resolvent energy of a graph G of order n is defined as ER(G) = ∑ n i=1 (n − λ i) −1 , where λ 1 ≥ λ 2 ≥ • • • ≥ λ n are the eigenvalues of G. Lower and upper bounds for the resolvent energy of a graph, which depend on some of the parameters n, λ 1 , λ n , det(R A (n)) = n ∏ i=1 1 n−λ i , are obtained.

Let G be a simple graph with vertex set V = V(G) = {v1, v2,..., vn} and edge set E = E(G). For vi ∈ V(G), by di = di(G) we denote the degree (number of neighbors) of the vertex vi. The second Zagreb index is defined as M2(G) = ∑ vivj∈E(G) didj. In this paper, we study the minimal and maximal second Zagreb index of trees with fixed diameter.

Let G = (V, E), V = {1, 2,. .. , n}, be a simple graph of order n and size m, without isolated vertices. Denote by ∆ = d 1 ≥ d 2 ≥ • • • ≥ d n = δ > 0, d i = d(i), a sequence of its vertex degrees. If vertices i and j are adjacent, we write i ∼ j. With T I we denote a topological index that can be represented as T I = T I(G) = ∑ i∼ j F(d i , d j), where F is an appropriately chosen function with the property F(x, y) = F(y, x). Randić degree-based adjacency matrix RA = (r i j) is defined as r i j = F(d i ,d j) √ d i d j if i ∼ j, and 0 otherwise. Denote by f i , i = 1, 2,. .. , n, the eigenvalues of RA. The Randić degree-based energy of graph could be defined as RE T I = RE T I (G) = ∑ n i=1 | f i |. Upper and lower bounds for RE T I are obtained.

Let G = (V, E), V = {1, 2,. .. , n}, be a simple graph of order n and size m, without isolated vertices. Denote by d 1 ≥ d 2 ≥ • • • ≥ d n > 0, d i = d(i), a sequence of its vertex degrees. If vertices i and j are adjacent, we write i ∼ j. With T I we denote a topological index that can be represented as T I = T I(G) = ∑ i∼ j F(d i , d j), where F is an appropriately chosen function with the property F(x, y) = F(y, x). Randić vertex-degree-based adjacency matrix RA = (r i j) is defined as r i j = F(d i ,d j) √ d i d j , if i ∼ j, and 0 otherwise. Denote by f 1 ≥ f 2 ≥ • • • ≥ f n the eigenvalues of RA. Upper and lower bounds for f i , i = 1, 2,. .. , n are obtained.