On the basis of graph G is adjointly unique if and only if is chromatically unique

it is mainly proved that for n≥1

a class of dense graphs T(1

1

n

4

1) is chromatically unique if and only if n≠1

4 and 7

by using the character

the algebraic properties of adjoint polynomial and distribution pattern of the minimum real roots of graph's

where T(1

1

n

4

1) denotes the graph with a path of length n+1

and from each end-vertex of this path its are drawn two paths of length 1

1 and two paths of length 4

1

respectively.