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Twin vertices in simple unweighted graphs are vertices that have the same neighbours, and, in the case of weighted graphs with possible loops, the corresponding incident edges have equal weights. In this paper, we explore the role of twin vertices in quantum state transfer. In particular, we provide characterizations of periodicity, perfect state transfer, and pretty good state transfer between twin vertices in a weighted graph with respect to its adjacency, Laplacian and signless Laplacian matrices. As an application, we provide characterizations of all simple unweighted double cones on regular graphs that exhibit periodicity, perfect state transfer, and pretty good state transfer. [ABSTRACT FROM AUTHOR]

We formalize the notion of a sedentary vertex and present a relaxation of the concept of a sedentary family of graphs introduced by Godsil (Linear Algebra Appl 614:356–375, 2021. https://doi.org/10.1016/j.laa.2020.08.027). We provide sufficient conditions for a given vertex in a graph to exhibit sedentariness. We also show that a vertex with at least two twins (vertices that share the same neighbours) is sedentary. We prove that there are infinitely many graphs containing strongly cospectral vertices that are sedentary, which reveals that, even though strong cospectrality is a necessary condition for pretty good state transfer, there are strongly cospectral vertices which resist high probability state transfer to other vertices. Moreover, we derive results about sedentariness in products of graphs which allow us to construct new sedentary families, such as Cartesian powers of complete graphs and stars. [ABSTRACT FROM AUTHOR]

The weighted graphs, where the edge weights are positive numbers, are considered. The authors obtain some lower bounds on the spectral radius and the Laplacian spectral radius of weighted graphs, and characterize the graphs for which the bounds are attained. Moreover, some known lower bounds on the spectral radius and the Laplacian spectral radius of unweighted graphs can be deduced from the bounds. [ABSTRACT FROM AUTHOR]

Let G be a simple connected graph of order n with degree sequence ( d, d, ..., d). Denote ( t) = Σ d, ( m) = ( t)/ d and ( N) = Σ ( t), where α is a real number. Denote by λ( G) and µ( G) the spectral radius of the adjacency matrix and the Laplacian matrix of G, respectively. In this paper, we present some upper and lower bounds of λ( G) and µ( G) in terms of ( t), ( m) and ( N). Furthermore, we also characterize some extreme graphs which attain these upper bounds. These results theoretically improve and generalize some known results. [ABSTRACT FROM AUTHOR]

Let G be a connected graph of order n. The diameter of a graph is the maximum distance between any two vertices of G. In this paper, we will give some bounds on the diameter of G in terms of eigenvalues of adjacency matrix and Laplacian matrix, respectively. [ABSTRACT FROM AUTHOR]