**MSC:**- 94A05 Communication theory, See also {60G35, 90B12}
- 05C35 Extremal problems, See also {90C35}

described in a group of individuals connected by a

communication network, whereby every node has a piece of

information and needs to transmit it to all the nodes in the

network. The networks are modelled by graphs, where the

vertices represent the nodes, and the edges the

communication links. In this paper, we concentrate on {\em

Minimum (Linear) Gossip Graphs} of even order, that is

graphs able to achieve gossiping in minimum time, and with a

minimum number of links. More precisely, we derive upper

bounds for their number of edges from a compounding method,

the {\em $k$-way split method}, previously introduced for

broadcasting by FARLEY. We show that this method can be

applied to gossiping in some cases, and that this

generalizes some compounding methods for gossip graphs given

given earlier by FERTIN. We also show that, when applicable,

this method gives the best known upper bound on the size of

Minimum Gossip Graphs in most cases, either improving or

matching them. Notably, we present for the first time two

infinite families of Gossip Graphs of respective degree

$\lceil\log_{2}(n)\rceil -3$ and $\lceil\log_{2}(n)\rceil

-4$. We also give some lower bounds on the number of edges

of Gossip Graphs which improve those given by FERTIN

before.\\

\indent Moreover, we show that the above compounding method

also applies for {\em Minimum Linear Gossip Graphs} (or

$MLGG$s) of even order, which corresponds to a variant of

gossiping where the time of information transmission between

two nodes depends on the amount of the information

exchanged. We also prove that this gives the best known

upper bounds for $G_{\beta,\tau}(n)$ - the size of a $MLGG$

of order $n$ - in most cases. In particular, we derive from

this method the exact value of $G_{\beta,\tau}(72)$, which

was previously unknown.