* Graphic: sterling and gilt yields http://bit.ly/2dgAXn1

* Graphic: World FX rates in 2017 http://tmsnrt.rs/2egbfVh

* Graphic: Trade-weighted sterling since Brexit vote http://tmsnrt.rs/2hwV9Hv

By Jemima Kelly

LONDON, July 4 (Reuters) - Sterling edged down on Tuesday after a survey of Britain's construction sector showed growth cooling in June, adding to signs that the economy might be struggling to gain momentum after a slow start to the year.

The pound, which at the start of the week was trading less than half a cent away from nine-month highs on bets the Bank of England could raise interest rates by the end of the year, dipped to a six-day low of $1.2918 after the data.

The purchasing managers' index (PMI) survey showed activity falling to 54.8 in June, still above the 50 level that denotes growth but down from May's 18-month high of 56.0 and slightly below the median forecast in a Reuters poll of economists.

The equivalent survey for the manufacturing sector, released on Monday, had shown activity grew much more slowly than forecast in June.

But MUFG currency economist Lee Hardman said these surveys should be taken "with a pinch of salt", as they reflected political uncertainty last month, when the ruling Conservative party unexpectedly lost its majority in parliamentary elections.

"We had heightened political uncertainty last month, and that could have weighed on business confidence," he said.

"That's potentially why the data is coming in weaker than expected, but I suppose one of the key tests going forward will be whether that loss of confidence will be sustained."

The BoE is pondering whether to raise interest rates for the first time in a decade, and is watching for signs that other areas of growth can offset a consumer spending slowdown, caused by a rise in inflation and a slowing in pay growth.

One of the BoE's interest rate-setters, Gertjan Vlieghe, said late on Monday he favoured keeping borrowing costs at their historic lows, despite a shift among some of his peers at the central bank in favour of a first hike in a decade.

But that seemed to have no impact on sterling, which recorded its best week in eight months last week after BoE Governor Mark Carney said a rate rise was likely to be needed as the economy came closer to full capacity, and that the Bank would debate this in the coming months.

"The BoE's lower tolerance for inflation means downside room for UK real yields is now much more limited. This means downside room for sterling is also more limited," wrote Nomura strategists in a morning note to clients.