The parties also congratulated Kumar for her "spirited" fight in the election.
CPI (M) general secretary Sitaram Yechury said on Twitter that the opposition's fight for a "plural and equitable" India will continue.
"Congratulations to Shri Ram Nath Kovind. Our fight for a plural, harmonious, equitable and just India continues.
"Congratulations to Smt Meira Kumar for a spirited fight. @meira_kumar," he said on the micro-blogging site.
CPI national secretary D Raja joined Yechury in congratulating the two and said Kovind should now act as a custodian of the Constitution.
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Kovind (71), who will be the second Dalit to hold the top constitutional position, garnered 65.65 percent of the votes in the Electoral College. He received 2930 votes with a value of 7,02,044, returning officer Anoop Mishra said.
Kovind will be the first BJP member to be elected president.
Kumar, also a Dalit, polled in 1844 votes with a value of 3,67,314.
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