Let G be a mixed graph. We discuss the relation between the second largest eigenvalue λ₂(G) and the second largest degree d₂(G), and present a sufficient condition for λ₂(G) ≥ d₂(G).

A graph is determined by its signless Laplacian spectrum if no other non-isomorphic graph has the same signless Laplacian spectrum (simply $G$ is $DQS$). Let $T(a,b,c)$ denote the $T$-shape tree obtained by identifying the end vertices of three paths $P_{a+2}$, $P_{b+2}$ and $P_{c+2}$. We prove that its all line graphs $ℒ\left(T\right(a,b,c\left)\right)$ except $ℒ\left(T\right(t,t,2t+1\left)\right)$ ($t\ge 1$) are $DQS$, and determine the graphs which have the same signless Laplacian spectrum as $ℒ\left(T\right(t,t,2t+1\left)\right)$. Let ${\mu }_1\left(G\right)$ be the maximum signless Laplacian eigenvalue of the graph $G$. We give the limit of ${\mu }_1\left(ℒ\left(T(a,b,c)\right)\right)$, too.

Let $A\left(G\right)$ be the adjacency matrix of $G$. The characteristic polynomial of the adjacency matrix $A$ is called the characteristic polynomial of the graph $G$ and is denoted by $\phi (G,\lambda )$ or simply $\phi \left(G\right)$. The spectrum of $G$ consists of the roots (together with their multiplicities) ${\lambda }_1\left(G\right)\ge {\lambda }_2\left(G\right)\ge ...\ge {\lambda }_n\left(G\right)$ of the equation $\phi (G,\lambda )=0$. The largest root ${\lambda }_1\left(G\right)$ is referred to as the spectral radius of $G$. A $‡$-shape is a tree with exactly two of its vertices having maximal degree 4. We will denote by $G(l_1,l_2,...,l_7)$$...$

We study the structure of path-like trees. In order to do this, we introduce a set of trees that we call expandable trees. In this paper we also generalize the concept of path-like trees and we call such generalization generalized path-like trees. As in the case of path-like trees, generalized path-like trees, have very nice labeling properties.

For a bipartite graph $G$ and a non-zero real $\alpha$, we give bounds for the sum of the $\alpha$th powers of the Laplacian eigenvalues of $G$ using the sum of the squares of degrees, from which lower and upper bounds for the incidence energy, and lower bounds for the Kirchhoff index and the Laplacian Estrada index are deduced.

The spectral radius of a graph is defined by that of its unoriented Laplacian matrix. In this paper, we determine the unicyclic graphs respectively with the third and the fourth largest spectral radius among all unicyclic graphs of given order.