The Gutman index (also known as Schultz index of the second kind) of a graph
$$G$$
is defined as
$$Gut(G)=\sum \nolimits _{u,v\in V(G)}d(u)d(v)d(u, v)$$
. A graph
$$G$$
is called a cactus if each block of
$$G$$
is either an edge or a cycle. Denote by
$$\mathcal {C}(n, k)$$
the set of connected cacti possessing
$$n$$
vertices and
$$k$$
cycles. In this paper, we give the first three smallest Gutman indices among graphs in
$$\mathcal {C}(n, k)$$
, the corresponding extremal grap...
