Mayumi and Mr.Akutagawa had a relationship of father and daughter and she loved her father dearly. Although, when he went on the brink of insanity her expression was shown to be very protective against her father to protect Koyori. Other than that, not much is known about their relationship although till the very end she did love her father as she along with her sister Koyori went into a coma.
Mayumi and Mrs. Akutagawa had the relationship of mother and daughter and her mother cared for her dearly as did Mayumi. Both Koyori and Mayumi seems to have taken after her looks. Mayumi loved her mother dearly as along when her mother and father's supposed death Koyori and Mayumi went into a coma.
Koyori is the younger twin-sister of Mayumi and they both love each other dearly. As mentioned by their father Koyori and Mayumi were born hand shakers and by the time they were infants become partners. Mayumi was willing to protect Koyori as much to her ability as she could.
Nagaoka is the current hand shaker partner of Mayumi and their relationship is not really much known to many degrees whether it's bad or good.