The classical Pauli group can be obtained as the central product of the dihedral group of 8 elements with the cyclic group of order 4. Inspired by this characterization, we introduce the notion of ...central product of Cayley graphs, which allows to regard the Cayley graph of a central product of groups as a quotient of the Cartesian product of the Cayley graphs of the factor groups. We focus our attention on the Cayley graph Cay(Pn,SPn) of the generalized Pauli group Pn on n-qubits; in fact, Pn may be decomposed as the central product of finite 2-groups, and a suitable choice of the generating set SPn allows us to recognize the structure of central product of graphs in Cay(Pn,SPn).
Using this approach, we are able to recursively construct the adjacency matrix of Cay(Pn,SPn) for each n≥1, and to explicitly describe its spectrum and the associated eigenvectors. It turns out that Cay(Pn,SPn) is a (3n+2)-regular bipartite graph on 4n+1 vertices, and it has integral spectrum. This is a highly nontrivial property if one considers that, by choosing as a generating set for P1 the three classical Pauli matrices, one gets the so-called Möbius-Kantor graph, belonging to the class of generalized Petersen graphs, whose spectrum is not integral.
We review a general class of models for self-organized dynamics based on alignment. The dynamics of such systems is governed solely by interactions among individuals or "agents," with the tendency to ...adjust to their "environmental averages." This, in turn, leads to the formation of clusters, e.g., colonies of ants, flocks of birds, parties of people, rendezvous in mobile networks, etc. A natural question which arises in this context is to ask when and how clusters emerge through the self-alignment of agents, and what types of "rules of engagement" influence the formation of such clusters. Of particular interest to us are cases in which the self-organized behavior tends to concentrate into one cluster, reflecting a consensus of opinions, flocking of birds, fish, or cells, rendezvous of mobile agents, and, in general, concentration of other traits intrinsic to the dynamics. Many standard models for self-organized dynamics in social, biological, and physical sciences assume that the intensity of alignment increases as agents get closer, reflecting a common tendency to align with those who think or act alike. Moreover, "similarity breeds connection" reflects our intuition that increasing the intensity of alignment as the difference of positions decreases is more likely to lead to a consensus. We argue here that the converse is true: when the dynamics is driven by local interactions, it is more likely to approach a consensus when the interactions among agents increase as a function of their difference in position. Heterophily, the tendency to bond more with those who are different rather than with those who are similar, plays a decisive role in the process of clustering. We point out that the number of clusters in heterophilious dynamics decreases as the heterophily dependence among agents increases. In particular, sufficiently strong heterophilious interactions enhance consensus.
In this article we establish relationships between Leavitt path algebras, talented monoids and the adjacency matrices of the underlying graphs. We show that indeed the adjacency matrix generates in ...some sense the group action on the generators of the talented monoid. With the help of this, we deduce a form of the aperiodicity index of a graph via the talented monoid. We classify hereditary and saturated subsets via the adjacency matrix. This then translates to a correspondence between the composition series of the talented monoid and the so-called matrix composition series of the adjacency matrix. In addition, we discuss the number of cycles in a graph. In particular, we give an equivalent characterization of acyclic graphs via the adjacency matrix, the talented monoid and the Leavitt path algebra. Finally, we compute the number of linearly independent paths of certain length in the Leavitt path algebra via adjacency matrices.
For a simple graph
G
, let
A
(
G
) be the adjacency matrix and
D
(
G
) be the diagonal matrix of the vertex degrees of graph
G
. For a real number
α
∈
0
,
1
, the generalized adjacency matrix
A
α
(
...G
)
is defined as
A
α
(
G
)
=
α
D
(
G
)
+
(
1
-
α
)
A
(
G
)
. The largest eigenvalue of the matrix
A
α
(
G
)
is the generalized adjacency spectral radius or the
A
α
-spectral radius of the graph
G
. In this paper, we obtain some new sharp lower and upper bounds for the generalized adjacency spectral radius of
G
, in terms of different parameters like vertex degrees, the maximum and the second maximum degrees, the number of vertices and the number of edges, etc, associated with the structure of graph
G
. The extremal graphs attaining these bounds are characterized. We show that our bounds improve some recent given bounds in the literature in some cases. Further, our results extend some known results for the adjacency and/or the signless Laplacian spectral radius of a graph
G
to a general setting.
The anti-adjacency matrix of a graph is constructed from the distance matrix of a graph by keeping each row and each column only the largest distances. This matrix can be interpreted as the opposite ...of the adjacency matrix, which is instead constructed from the distance matrix of a graph by keeping in each row and each column only the distances equal to 1. The (anti-)adjacency eigenvalues of a graph are those of its (anti-)adjacency matrix. Employing a novel technique introduced by Haemers (2019) 9, we characterize all connected graphs with exactly one positive anti-adjacency eigenvalue, which is an analog of Smith's classical result that a connected graph has exactly one positive adjacency eigenvalue iff it is a complete multipartite graph. On this basis, we identify the connected graphs with all but at most two anti-adjacency eigenvalues equal to −2 and 0. Moreover, for the anti-adjacency matrix we determine the HL-index of graphs with exactly one positive anti-adjacency eigenvalue, where the HL-index measures how large in absolute value may be the median eigenvalues of a graph. We finally propose some problems for further study.
Let G be a simple connected graph and A(G) be its adjacency matrix. The terms singularity, eigenvalues, and characteristic polynomial of G mean those of A(G). A nonsingular graph G is said to have ...the reciprocal eigenvalue property if the reciprocal of each eigenvalue of G is also an eigenvalue. A graph G (possibly singular) is said to have the weak reciprocal eigenvalue property if the reciprocal of each nonzero eigenvalue of it is also an eigenvalue. In Barik et al. (2022) 3, the authors proved that there is no nontrivial tree with the weak reciprocal eigenvalue property and posed the following question: “Does there exist a nontrivial graph with the weak reciprocal eigenvalue property?” Suppose that G is singular and the characteristic polynomial of G is xn−k(xk+a1xk−1+⋯+ak). Assume that A(G) has rank k, so that ak≠0. Can we ever have |ak|=1? The answer turns out to be negative. As an application, we settle the question posed in Barik et al. (2022) 3. Another similar application is also mentioned. It is natural to wonder, “Does there exist a nontrivial, simple, connected weighted graph with the weak reciprocal eigenvalue property?” We provide a class of such graphs. Furthermore, we extend our results to weighted graphs.
In monocular 3D human pose estimation, modeling the temporal relation of human joints is crucial for prediction accuracy. Currently, most methods utilize transformer to model the temporal relation ...among joints. However, existing transformer-based methods have limitations. The temporal adjacency matrix utilized within the self-attention of the temporal transformer inaccurately models the temporal relationships between frames, particularly in cases where distinct motions exhibit significant correlation despite having different physical interpretations and large temporal spans. To address this issue, we construct an artificial temporal adjacency matrix based on input data and introduce a temporal adjacency matrix hybrid module to blend this matrix with the model’s inherent temporal adjacency matrix, resulting in a novel composite temporal adjacency matrix. Through extensive experiments on Human3.6M and MPI-INF-3DHP datasets using state-of-the-art methods as benchmarks, our proposed method demonstrates a maximum improvement of up to 5.6% compared to the original approach.
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