Following an idea of spectral graph theory, we give several characterizations of finite one-peak posets I with almost P-critical quadratic Tits form (q) over barI : Z(I) ->Z by applying linear algebra methods, combinatorial algorithms, an old idea of S.A. Ovsienko and recent results by V.M. Bondarenko and M.V. Stepochkina. Our study is mainly motivated by an important application of the quadratic Tits form of a poset in constructing linear algebra invariants that measure a geometric complexity of Nazarova-Roiter matrix problems over a field K. In particular, our study is inspired by a well-known result of Ju.A. Drozd explained in Introduction. One of our main results asserts that, given a finite poset I = {1, ... , n,*} = T boolean OR {*1 such that * is its unique maximal element and (q) over barI is almost P-critical, we have: (i) the form (q) over capI is non-negative and the subgroup Ker (q) over capI := {v is an element of Z(n+1); (q) over capI(v) = 0} of Z(I) = Z(n+1) is generated by a vector h = (h(1), ..., h(*)), with h1 not equal 0, ... , h(n) not equal 0 , (ii) 5Z of the Euclidean diagram Delta(I,) (iv) there is a Z-invertible matrix C such that (C) over cap (tr)(I) = C-tr. (C) over cap (I).C and C-2 = E, where (C) over cap (I) is an element of Mn+1(Z) is the Tits matrix of I, and matrix of I, and (v)I is one of the posets L1, ..., L-132 listed in the paper. In particular, we get in an alternative way the list of P-critical posets T (in the form T = I \ {*}) presented by V.M. Bondarenko and M.V. Stepochkina [6], and we determine their Coxeter-Euclidean types. There are 115 one-peak posets I is an element of {L-1, ... , L-115}, with (C) over cap (I) P-critical, and 17 exceptional posets I is an element of {L-116, ..., L-132}, with (C) over cap (I) almost P-critical, but not P-critical. Our lists are obtained by using computer algebra tools; mainly symbolic computation in Maple and numeric computation in C#. (C) 2013 Elsevier Inc. All rights reserved.