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arXiv:2610.13349

A General Method for Extending Isoscattering Non-Isospectral Quantum Graphs

Omer Farooq, Michal Lawniczak, Leszek Sirko

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

Abstract

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 $Γ_1$ and $Γ_2$, 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 $k_n$, 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.

Comments
18 pages, 4 Figures

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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.
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 scattering amplitude in simple quantum graphs is a well-known process which may be highly complex. In this work, motivated by the Shannon entropy, we propose a methodology that associates a graph with a scattering entropy, which we call the average scattering entropy. It is defined by taking into account the period of the scattering amplitude which we calculate using the Green's function procedure. We first describe the methodology on general grounds and then exemplify our findings considering several distinct groups of graphs. We go on and investigate other possibilities, one that contains groups of graphs with the same number of vertices, with the same degree, and the same number of edges, with the same length, but with distinct topologies and with different entropies. Another possibility we investigate contains graphs of the fishbone type, where the scattering entropy depends on the boundary conditions on the vertices of degree 1, with the corresponding values decreasing and saturating very rapidly as we increase the number of elementary structures in the graphs.
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.
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 discuss a particular kind of quantum walk on a general graph. We affix two semi-infinite lines to a general finite graph, which we call tails. On the tails, the particle making the walk simply advances one unit at each time step, so that its behavior there is analogous to free propagation We are interested in how many steps it will take the particle, starting on one tail and propagating through the graph (where its propagation is not free), to emerge onto the other tail. The probability to make such a walk in n steps and the hitting time for such a walk can be expressed in terms of the transmission amplitude for the graph, which is one element of its S matrix. Demonstrating this necessitates a study of the analyticity properties of the transmission and reflection amplitudes of a graph. We show that the graph can have bound states that cannot be accessed by a particle entering the graph from one of its tails. Time-reversal invariance of a quantum walk is defined and used to show that the transmission amplitudes for the particle entering the graph from different directions are the same if the walk is time-reversal invariant.
We study quantum walks on general graphs from the point of view of scattering theory. For a general finite graph we choose two vertices and attach one half line to each. We are interested in walks that proceed from one half line, through the graph, to the other. The particle propagates freely on the half lines but is scattered at each vertex in the original graph. The probability of starting on one line and reaching the other after n steps can be expressed in terms of the transmission amplitude for the graph. An example is presented. Classical random walks on graphs can be used to construct algorithms that solve 2-SAT, graph connectivity problems, and for finding satsifying assignments for Boolean functions. A hope is that recently defined quantum walks will prove similarly useful in the development of quantum algorithms. In fact, it has recently been shown that it is possible to use a quantum walk to perform a search on the hypercube faster than can be done classically [1]. In this problem