Quantum state transfer between twins in weighted graphs

Sedentariness in quantum walks

“…To prove (2), note that the condition that θ is a simple eigenvalue of M implies that T " tu, vu and u and v are strongly cospectral [Mon22, Corollary 3.14]. Combining this with Corollary 12 and the characterizations of PST and PGST between twin vertices (see Theorems 11 and 14 in [KMP23]) yields the desired results in (2). The last statement in (2) follows from Lemma 8.…”

“…In (1b), we have n " pk ´2qm `2 and we may write X " K 2 _ Y, where Y " Ž k´2 O m is a pn ´m ´2q-regular graph on n ´2 vertices. Invoking Corollary 12 and Theorems 12(1) and 14(2b) in [KMP23] yields (1b). Now, (1ci) is immediate from Theorem 15(1,2).…”

“…Let d " 2s so that n " 3s 2 2 . In this case, ν 2 pdq ą ν 2 psq, and so PST occurs between the apexes of Y at some time τ by [KMP23,Theorem 11(1)]. This implies that Re `UApYq pτq v,v ˘" 0, and so by Theorem 13(2), we get that pu, vq is not sedentary in K 2 _ Y for any vertex u of K 2 .…”

Quantum walks on blow-up graphs

“…and G is obtained by adding a vertex u / ∈ V(H) that is adjacent to all vertices of H. We call u the apex of G. It can be checked that if H is a k-regular graph on m vertices and G is a cone K 1 + H, then G is a graph obtained by adding two vertices u and w that are adjacent to all vertices of We call u and w apexes of the double cone. Periodicity, PST and PGST between apexes of disconnected double cones over regular graphs were characterized by Kirkland et al[KMP23].The following is immediate from theorems 11(1) and 14(2a) in[KMP23].…”

“…Thus, blow-ups have the advantage of producing pairs of vertices that are promising for high probability state transfer. Lastly, the two copies of a vertex in G are twins in 2 ⊎ G. This allows us to utilize some of the recent work on quantum state transfer between twins [KMP23,Mon23a,Pal22]. In particular, we give characterizations of strong cospectrality, PST, PGST and periodicity in blow-up graphs (see theorems 2-5).…”