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Unimodality of the normalized coefficients of the characteristic polynomial of distance matrices of trees are known and bounds on the location of its peak (the largest coefficient) are also known. Recently, an extension of these results to distance matrices of block graphs was given. In this work, we extend these results to two additional distance-type matrices associated with trees: the Min-4PC matrix and the 2-Steiner distance matrix. We show that the sequences of coefficients of the characteristic polynomials of these matrices are both unimodal and log-concave. Moreover, we find the peak location for the coefficients of the characteristic polynomials of the Min-4PC matrix of any tree on $n$ vertices. Further, we show that the Min-4PC matrix of any tree on $n$ vertices is isometrically embeddable in $\mathbb{R}^{n-1}$ equipped with the $\ell_1$ norm.

For a connected graph $G$, let $A(G)$ be the adjacency matrix of $G$ and $D(G)$ be the diagonal matrix of the degrees of the vertices in $G$. The $A_{\alpha}$-matrix of $G$ is defined as \begin{align*} A_\alpha (G) = \alpha D(G) + (1-\alpha) A(G) \quad \text{for any $\alpha \in [0,1]$}. \end{align*} The largest eigenvalue of $A_{\alpha}(G)$ is called the $A_{\alpha}$-spectral radius of $G$. In this article, we characterize the graphs with maximum $A_{\alpha}$-spectral radius among the class of unicyclic and bicyclic graphs of order $n$ with fixed girth $g$. Also, we identify the unique graphs with maximum $A_{\alpha}$-spectral radius among the class of unicyclic and bicyclic graphs of order $n$ with $k$ pendant vertices., Comment: 16 pages

In this article, we show that the generalized tree shift operation increases the distance spectral radius, distance signless Laplacian spectral radius, and the $D_\alpha$-spectral radius of complements of trees. As a consequence of this result, we correct an ambiguity in the proofs of some of the known results.

The eccentricity matrix of a connected graph $G$, denoted by $\mathcal{E}(G)$, is obtained from the distance matrix of $G$ by keeping the largest nonzero entries in each row and each column, and leaving zeros in the remaining ones. The $\mathcal{E}$-eigenvalues of $G$ are the eigenvalues of $\mathcal{E}(G)$, in which the largest one is the $\mathcal{E}$-spectral radius of $G$. The $\mathcal{E}$-energy of $G$ is the sum of the absolute values of all $\mathcal{E}$-eigenvalues of $G$. In this article, we study some of the extremal problems for eccentricity matrices of complements of trees and characterize the extremal graphs. First, we determine the unique tree whose complement has minimum (respectively, maximum) $\mathcal{E}$-spectral radius among the complements of trees. Then, we prove that the $\mathcal{E}$-eigenvalues of the complement of a tree are symmetric about the origin. As a consequence of these results, we characterize the trees whose complement has minimum (respectively, maximum) least $\mathcal{E}$-eigenvalues among the complements of trees. Finally, we discuss the extremal problems for the second largest $\mathcal{E}$-eigenvalue and the $\mathcal{E}$-energy of complements of trees and characterize the extremal graphs. As an application, we obtain a Nordhaus-Gaddum type lower bounds for the second largest $\mathcal{E}$-eigenvalue and $\mathcal{E}$-energy of a tree and its complement., Comment: 18 pages. Preliminary version

The eccentricity matrix of a connected graph $G$, denoted by $\mathcal{E}(G)$, is obtained from the distance matrix of $G$ by keeping the largest nonzero entries in each row and each column and leaving zeros in the remaining ones. The eigenvalues of $\mathcal{E}(G)$ are the $\mathcal{E}$-eigenvalues of $G$. The eccentricity energy (or the $\mathcal{E}$-energy) of $G$ is the sum of the absolute values of all $\mathcal{E}$-eigenvalues of $G$. In this article, we determine the unique tree with the minimum second largest $\mathcal{E}$-eigenvalue among all trees on $n$ vertices other than the star. Also, we characterize the trees with minimum $\mathcal{E}$-energy among all trees on $n$ vertices., Comment: 18 Pages

Let $T$ be a tree on $n$ vertices whose edge weights are positive definite matrices of order $s$. The squared distance matrix of $T$, denoted by $\Delta$, is the $ns \times ns$ block matrix with $\Delta_{ij}=d(i,j)^2$, where $d(i,j)$ is the sum of the weights of the edges in the unique $(i,j)$-path. In this article, we obtain a formula for the determinant of $\Delta$ and find ${\Delta}^{-1}$ under some conditions., Comment: Preliminary version. Comments are welcome. 17 pages

The \textit{eccentricity matrix} $\mathcal{E}(G)$ of a connected graph $G$ is obtained from the distance matrix of $G$ by keeping the largest non-zero entries in each row and each column, and leaving zeros in the remaining ones. The eigenvalues of $\mathcal{E}(G)$ are the \textit{$\mathcal{E}$-eigenvalues} of $G$. In this article, we find the inertia of the eccentricity matrices of trees. Interestingly, any tree on more than $4$ vertices with odd diameter has two positive and two negative $\mathcal{E}$-eigenvalues (irrespective of the structure of the tree). A tree with even diameter has the same number of positive and negative $\mathcal{E}$-eigenvalues, which is equal to the number of 'diametrically distinguished' vertices (see Definition 3.1). Besides we prove that the spectrum of the eccentricity matrix of a tree is symmetric with respect to the origin if and only if the tree has odd diameter. As an application, we characterize the trees with three distinct $\mathcal{E}$-eigenvalues., Comment: Some of the typos are fixed. Comments are welcome!

The eccentricity matrix $\varepsilon(G)$ of a graph $G$ is obtained from the distance matrix of $G$ by retaining the largest distances in each row and each column, and leaving zeros in the remaining ones. The eccentricity energy of $G$ is sum of the absolute values of the eigenvalues of $\varepsilon(G)$. Although the eccentricity matrices of graphs are closely related to the distance matrices of graphs, a number of properties of eccentricity matrices are substantially different from those of the distance matrices. The change in eccentricity energy of a graph due to an edge deletion is one such property. In this article, we give examples of graphs for which the eccentricity energy increase (resp., decrease) but the distance energy decrease (resp., increase) due to an edge deletion. Also, we prove that the eccentricity energy of the complete $k$-partite graph $K_{n_1,\hdots,n_k}$ with $k\geq 2$ and $ n_i\geq 2$, increases due to an edge deletion., Comment: Preliminary version. 11 pages

Let $G$ be a simple, connected graph and let $A(G)$ be the adjacency matrix of $G$. If $D(G)$ is the diagonal matrix of the vertex degrees of $G$, then for every real $\alpha \in [0,1]$, the matrix $A_{\alpha}(G)$ is defined as $$A_{\alpha}(G) = \alpha D(G) + (1- \alpha) A(G).$$ The eigenvalues of the matrix $A_{\alpha}(G)$ form the $A_{\alpha}$-spectrum of $G$. Let $G_1 \dot{\vee} G_2$, $G_1 \underline{\vee} G_2$, $G_1 \langle \textrm{v} \rangle G_2$ and $G_1 \langle \textrm{e} \rangle G_2$ denote the subdivision-vertex join, subdivision-edge join, $R$-vertex join and $R$-edge join of two graphs $G_1$ and $G_2$, respectively. In this paper, we compute the $A_{\alpha}$-spectra of $G_1 \dot{\vee} G_2$, $G_1 \underline{\vee} G_2$, $G_1 \langle \textrm{v} \rangle G_2$ and $G_1 \langle \textrm{e} \rangle G_2$ for a regular graph $G_1$ and an arbitrary graph $G_2$ in terms of their $A_{\alpha}$-eigenvalues. As an application of these results, we construct infinitely many pairs of $A_{\alpha}$-cospectral graphs.

The eccentricity matrix $\varepsilon(G)$ of a graph $G$ is obtained from the distance matrix by retaining the eccentricities (the largest distance) in each row and each column. In this paper, we give a characterization of the star graph, among the trees, in terms of invertibility of the associated eccentricity matrix. The largest eigenvalue of $\varepsilon(G)$ is called the $\varepsilon$-spectral radius, and the eccentricity energy (or the $\varepsilon$-energy) of $G$ is the sum of the absolute values of the eigenvalues of $\varepsilon(G)$. We establish some bounds for the $\varepsilon$-spectral radius and characterize the extreme graphs. Two graphs are said to be $\varepsilon$-equienergetic if they have the same $\varepsilon$-energy. For any $n \geq 5$, we construct a pair of $\varepsilon$-equienergetic graphs on $n$ vertices, which are not $\varepsilon$-cospectral., Comment: 11 Pages

The eccentricity matrix of a connected graph $G$ is obtained from the distance matrix of $G$ by retaining the largest distances in each row and each column, and setting the remaining entries as $0$. In this article, a conjecture about the least eigenvalue of eccentricity matrices of trees, presented in the article [Jianfeng Wang, Mei Lu, Francesco Belardo, Milan Randic. The anti-adjacency matrix of a graph: Eccentricity matrix. Discrete Applied Mathematics, 251: 299-309, 2018.], is solved affirmatively. In addition to this, the spectra and the inertia of eccentricity matrices of various classes of graphs are investigated., Comment: Comments are welcome!