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arXiv:2610.13512

Construction of isoscattering non-isospectral quantum graphs using the concept of a germ graph

Omer Farooq, Michał Ławniczak, Leszek Sirko

quant-phSubmitted Oct 9, 2026• Announced Oct 12, 2026
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Abstract

The recently developed concept of germ graphs (P. Kurasov {\it et al.}, Phys. Rev. Research {\bf 7}, L022071 (2025)) is employed to construct families of isocattering, non-isospectral graphs that exhibit dissipation. More specifically, a new pair of isoscattering non-isospectral graphs is constructed by connecting the existing isoscattering non-isospectral graphs, so called attachments, to the different contact vertices of the germ graph. The existing isoscattering non-isospectral graphs contained a two-loop graph labelled $Γ_{A2}$ and a three-loop graph labelled $Γ_{A3}$. These graphs have an equal total metric length, but different topologies. The resulting graphs $Γ_3$ and $Γ_4$ are verified as non-isospectral using their Euler characteristics, while their isoscattering properties are investigated by calculating the one-port scattering matrices numerically.

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The recently developed concept of germ graphs (P. Kurasov {\it et al.}, Phys. Rev. Research {\bf 7}, L022071 (2025)) is employed to construct families of isocattering, non-isospectral graphs that exhibit dissipation. More specifically, a new pair of isoscattering non-isospectral graphs is constructed by connecting the existing isoscattering non-isospectral graphs, so called attachments, to the different contact vertices of the germ graph. The existing isoscattering non-isospectral graphs contained a two-loop graph labelled ΓA2Γ_{A2}ΓA2​ and a three-loop graph labelled ΓA3Γ_{A3}ΓA3​. These graphs have an equal total metric length, but different topologies. The resulting graphs Γ3Γ_3Γ3​ and Γ4Γ_4Γ4​ are verified as non-isospectral using their Euler characteristics, while their isoscattering properties are investigated by calculating the one-port scattering matrices numerically.
The present paper sets out a general method for extending the isoscattering, non-isospectral quantum graphs. The methodology is illustrated by considering two graphs, denoted by Γ1Γ_1Γ1​ and Γ2Γ_2Γ2​, which have the same total metric length but different topologies. It is demonstrated that, despite the fact that their scattering matrices are identical, their eigenvalues, denoted by knk_nkn​, and Euler characteristics differ from one another. The next step in the process is to extend both graphs by affixing the same star graph to each of them. Utilising the Titchmarsh-Weyl M-function formalism, it is proven that the newly constructed, extended graphs are also isoscattering and non-isospectral.
One of the most important characteristics of a quantum graph is the average density of resonances, ρ=(L/π), where L denotes the length of the graph. This is a very robust measure. It does not depend on the number of vertices in a graph and holds also for most of the boundary conditions at the vertices. Graphs obeying this characteristic are called Weyl graphs. Using microwave networks that simulate quantum graphs we show that there exist graphs that do not adhere to this characteristic. Such graphs are called non-Weyl graphs. For standard coupling conditions we demonstrate that the transition from a Weyl graph to a non-Weyl graph occurs if we introduce a balanced vertex. A vertex of a graph is called balanced if the numbers of infinite leads and internal edges meeting at a vertex are the same. Our experimental results confirm the theoretical predictions of [E. B. Davies and A. Pushnitski, Analysis and PDE 4, 729 (2011)] and are in excellent agreement with the numerical calculations yielding the resonances of the networks.
We give a new formula for computing the isospectral reduction of a matrix (and graph) down to a submatrix (or subgraph). Using this, we generalize the notion of isospectral reductions. In addition, we give a procedure for constructing a matrix whose isospectral reduction down to a submatrix is given. We also prove that the isospectral reduction completely determines the restriction of the quantum walk transition matrix to a subset. Using these, we construct new families of simple graphs exhibiting perfect quantum state transfer.
We investigate the graph isomorphism (GI) in some cospectral networks. Two graph are isomorphic when they are related to each other by a relabeling of the graph vertices. We want to investigate the GI in two scalable (n + 2)-regular graphs G4(n; n + 2) and G5(n; n + 2), analytically by using the multiparticle quantum walk. These two graphs are a pair of non-isomorphic connected cospectral regular graphs for any positive integer n. In order to investigation GI in these two graphs, we rewrite the adjacency matrices of graphs in the antisymmetric fermionic basis and show that they are different for thesepairs of graphs. So the multiparticle quantum walk is able to distinguish pairs of non- isomorph graphs. Also we construct two new graphs T4(n; n + 2) and T5(n; n + 2) and repeat the same process of G4 and G5 to study the GI problem by using multiparticle quantum walk. Then we study GI by using the entanglement entropy. To this aim, we calculate entanglement entropy between two parts of network. In our model the nodes are considered as identical quantum harmonic oscillators. The entanglement entropy between two special parts of G4(n; n+2) and G5(n; n+2) are calculated analytically. It is shown that the entanglement entropy can distinguish pairs of non-isomorphic cospectral graphs too.
Abstract The study of multipartite entanglement is much less developed than the bipartite scenario. Recently, a string of results have proposed using tools from topological data analysis to attach topological quantities to multipartite states. However, these quantities are not directly connected to concrete information processing tasks making their interpretations vague. We take the first steps in connecting these abstract topological quantities to operational interpretations of entanglement in two scenarios. The first is we provide a bound on the average distillable entanglement, which we develop as a generalization of the Meyer–Wallach entanglement measure studied by Scott (2004 Phys. Rev. A 69 052330), via the integrated Euler characteristic (IEC) defined by Hamilton and Leditzky. This allows us to connect the distance of an error correcting code to the IEC. The second is we provide a characterization of a class of graph states containing the GHZ state via the birth and death times of the connected components and one-dimensional cycles of the entanglement complex. In other words, the entanglement distance behavior of the first Betti number β 1 ( ε ) allows us to determine if a state is locally equivalent to a GHZ state, potentially providing new verification schemes for highly entangled states.