Yuan, Shao and Liu proved  that the H-shape tree $H'_n =P_{1,2;n-3}^{1,n-6}$ minimizes the spectral radius among all graphs with order $n\geqslant 9$ and diameter $n-4$. In this paper, we achieve the spectral characterization of all graphs in the set $\mathscr{H}' = \{ H'_n\}_{n\geqslant 8}$. More precisely we show that $H'_n$ is determined by its spectrum if and only if $n \neq 8, 9,12$, and detect all cospectral mates of $H'_8$, $H'_9$ and $H'_{12}$. Divisibility between characteristic polynomials of graphs turns out to be an important tool to reach our goals.

In this paper we give more results about inner radius spectrum of operators on Hilbert spaces with several examples. Also, we established an inequality for inner radius spectrum of a positive operator matrix and its minimum moduli block matrix.

In this paper a new quantity for real tensors, the sign-real spectral radius, is defi ned and investigated. Various characterizations, bounds and some properties are derived. In certain aspects our quantity shows similar behavior to the spectral radius of a nonnegative tensor. In fact, we generalize the Perron Frobenius theorem for nonnegative tensors to the class of real tensors.

A characterization of nonlinear spectral radius preserving maps is obtained for the usual and triple products of nonnegative matrices.

Let D be the diameter and dG(vi, vj ) be the distance between the vertices vi and vj of a connected graph G. The complementary distance matrix of a graph G is CD(G) = [cdij ] in which cdij = 1 +D − dG(vi, vj ) if i 6= j and cdij = 0 if i = j. The complementary transmission CTG(v) of a vertex v is defined as CTG(v) = Pu2V (G)[1+D− dG(u, v)]. Let CT(G) = diag[CTG(v1), CTG(v2), . . ., CTG(vn)]. The complementary distance signless Laplacian matrix of G is CDL+(G) = CT(G) + CD(G). In this paper, we obtain the bounds for the largest eigenvalue of CDL+(G). Further we determine Nordhaus-Gaddum type results for the largest eigenvalue. We also establish some bounds for the complementary distance signless Laplacian energy.