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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]

Let R be a commutative ring with all non-zero zero divisors Z ∗ (R) of R . Then Γ (R) is said to be a zero divisor graph if and only if a · b = 0 where a , b ∈ V (Γ (R)) = Z ∗ (R) and (a , b) ∈ E (Γ (R)) . Graph energy E (Γ (R)) is defined as the sum of the absolute eigenvalues of the adjacency matrix of Γ (R) , then E (Γ (R)) = ∑ i = 1 n | λ i | . A topological index is a numeric quantity associated with a chemical structure that attempts to link the chemical structure to various physicochemical properties, chemical reactivity, or biological activity. This paper discusses the graph energy and various topological indices of zero divisor graph associated with the commutative ring. [ABSTRACT FROM AUTHOR]

Let G be a graph with vertex set { v 1 , v 2 , ... , v n } and let d i denote the degree of vertex v i . The geometric–arithmetic matrix GA (G) of G is indexed by the vertices of G, whose (i, j) -th entry is 2 d i d j d i + d j if the vertices i and j are adjacent and 0 otherwise. The multi-set of eigenvalues of GA (G) is known as the geometric–arithmetic spectrum of G. In this article, we obtain geometric–arithmetic spectra of various families of graphs and characterize the connected graphs with two and three distinct GA eigenvalues. We obtain several upper and lower bounds for the geometric arithmetic energy and characterize the graphs attaining such bounds. [ABSTRACT FROM AUTHOR]

We present a method to derive the complete spectrum of the lift Γ α of a base digraph Γ , with voltage assignment α on a (finite) group G. The method is based on assigning to Γ a quotient-like matrix whose entries are elements of the group algebra C [ G ] , which fully represents Γ α . This allows us to derive the eigenvectors and eigenvalues of the lift in terms of those of the base digraph and the irreducible characters of G. Thus, our main theorem generalizes some previous results of Lovász and Babai concerning the spectra of Cayley digraphs. [ABSTRACT FROM AUTHOR]

A graph is reflexive if the second largest eigenvalue of its adjacency matrix is less than or equal to 2. In this paper, we characterize trees whose line graphs are reflexive. It turns out that these trees can be of arbitrary order-they can have either a unique vertex of arbitrary degree or pendant paths of arbitrary lengths, or both. Since the reflexive line graphs are Salem graphs, we also relate some of our results to the Salem (graph) numbers. [ABSTRACT FROM AUTHOR]

Let $$\fancyscript{B}(n,r)$$ be the set of all bicyclic graphs with $$n$$ vertices and $$r$$ cut edges. In this paper we determine the unique graph with maximal adjacency spectral radius or signless Laplacian spectral radius among all graphs in $$\fancyscript{B}(n,r)$$ . [ABSTRACT FROM AUTHOR]

An elegant full characterization of reflexive line graphs of trees has proved to be quite difficult task. This paper tries to shed some more light on known results about such graphs by providing more numerical details regarding their structural composition. The paper also presents numerous results and ideas on the topic, as well as some observations with respect to the connection with Salem numbers. [ABSTRACT FROM AUTHOR]