Let $T$ be a tree of order $n$ and $S_2(T)$ the sum two largest Laplacian eigenvalues $T$. Fritscher et al. proved that for any $n$, $S_2(T) \leq n+2-\frac{2}{n}$. Guan determined with maximum among all trees $n$. In this paper, we characterize \geq n+1$ except some trees. Moreover, also determine first $\lfloor\frac{n-2}{2}\rfloor$ according to their $S_2(T)$. This extends result

I n 1912 Hermann Weyl [W] posed the following problem: given the eigenvalues of two n× n Hermitian matrices A and B, how does one determine all the possible sets of eigenvalues of the sum A + B? When n = 1, the eigenvalue of A + B is of course just the sum of the eigenvalue of A and the eigenvalue of B, but the answer is more complicated in higher dimensions. Weyl’s partial answers to this prob...

the energy of a graph is equal to the sum of the absolute values of its eigenvalues. two graphs of the same order are said to be equienergetic if their energies are equal. we point out the following two open problems for equienergetic graphs. (1) although it is known that there are numerous pairs of equienergetic, non-cospectral trees, it is not known how to systematically construct any such pa...

let $g=(v,e)$ be a connected graph. the eccentric connectivity index of $g$, $xi^{c}(g)$, is defined as $xi^{c}(g)=sum_{vin v(g)}deg(v)ec(v)$, where $deg(v)$ is the degree of a vertex $v$ and $ec(v)$ is its eccentricity. the eccentric distance sum of $g$ is defined as $xi^{d}(g)=sum_{vin v(g)}ec(v)d(v)$, where $d(v)=sum_{uin v(g)}d_{g}(u,v)$ and $d_{g}(u,v)$ is the distance between $u$ and $v$ ...

The D-eigenvalues of a graph G are the eigenvalues of its distance matrix D , and the D-energy ED(G) is the sum of the absolute values of its D-eigenvalues. Two graphs are said to be D-equienergetic if they have the same D-energy. In this note we obtain bounds for the distance spectral radius and D-energy of graphs of diameter 2. Pairs of equiregular D-equienergetic graphs of diameter 2, on p =...

The eigenvalues of a digraph are the eigenvalues of its adjacency matrix. The sum of the absolute values of the real part of the eigenvalues is called the energy of the digraph. The extremal energy of bicyclic digraphs with vertex-disjoint directed cycles is known. In this paper, we consider a class of bicyclic digraphs with exactly two linear subdigraphs of equal length. We find the minimal an...

the paper presents a mathematical base modeling combined to modified-winding -function-approach (mwfa) for eccentricity fault detection of a round-rotor synchronous machine. for this aim, a 6-pole machine is considered, and the machine inductances are computed by mwfa in healthy and also under eccentricity fault. a numerical discrete-time method has been proposed to machine modeling in voltage-...