Some transformations that preserve sgn(λ 2 ≤ r) Bojana Mihailović, Marija Rašajski Elektrotehnički fakultet, Beograd SGA 2016, Beograd, Serbia

Introduction Simple, nonoriented graphs (without loops or multiple edges) with symmetric adjacency matrix Real eigenvalues of its characteristic polynomial P G (λ) are λ 1,λ 2,..., λ n (λ 1 ≥ λ 2 ≥... ≥ λ n ) Interlacing theorem: Let A be a symmetric matrix with eigenvalues λ 1, λ 2,..., λ n and B one of its principal submatrices with eigenvalues μ 1, μ 2,..., μ m. Then the inequalities λ n-m+i ≤ ≤ μ i ≤ λ i (i=1,...,m) hold. λ i ≤ r is a hereditary property

Scwenk’s lemmas: Given a graph G, let C(v) (C(uv)) denote the set of all cycles containing a vertex v and an edge uv of G, respectively. Then: where Adj(v) denotes the set of neighbors of v, φ(v) (φ(uv)) the sets of cycles that contain v (uv), while G-V(C) is the graph obtained from G by removing the vertices belonging to the cycle C. Let G 1 and G 2 be two rooted graphs with the roots u 1 and u 2, respectively, and let G 1 ∙G 2 be their coalescence. Then:

Let G be a graph with a cut-vertex u. The components G 1,…,G n of the graph G-u are connected and 1)If at most one of the components has index r, and for the rest of them indices are less than r, then 2)If at least two of the components have index r, and for the rest of them indices are not greater then r, then 3)If only one of the components has index greater than r, and at least one of the remaining components has index r, and for the rest of them indices are less than r, then 4)If at least two of the components have indices greater than r, then Generalized RS-theorem

α - transformation Let V be the family of connected graphs G=G(X,A,B) with at least two bridges, whose subgraphs X, A and B are rooted graphs, with the roots x, a and b, respectively, where λ 1 (X)=r (r > 0). α: V →V, α(G 1 )=G 2 where G 1 =G(X,A,B), G 2 =G(X,A∙B,b) → Theorem: For G 1 =G(X,A,B), sgn(λ 2 (G 1 )-r)=sgn(λ 2 (α(G 1 ))-r) holds.

Proof: If λ 1 (A-a) ≥ r or λ 1 (B-b) ≥ r, sgn(λ 2 (G 1 )-r)=sgn(λ 2 (α(G 1 ))-r)=1 holds (GRS- theorem). Suppose that λ 1 (A-a) < r and λ 1 (B-b) < r. Then λ 3 (G 1 )<r and λ 3 (β(G 1 ))< r holds (GRS-theorem). Therefore, sgnP G1 =sgn(λ 2 (G 1 )-r) i sgnP α(G) =sgn(λ 2 (α(G 1 ))-r). Using Schwenk’s lemma we get: Because of P S =0, we get P G1 =P α(G1), so sgn(λ 2 (G 1 )-r)=sgn(λ 2 (α(G 1 ))-r) holds.

β - transformation Let U be the family of connected graphs G=G(X,x,A,y,B) with at least two bridges, whose subgraphs A and B are rooted graphs, with the roots a and b, while its subgraph X, such that λ 1 (X)=r (r > 0), has two particular vertices, x and y - the endpoints of the bridges xa and yb. β: V →V, β(G 1 )=G 2 where G 1 =G(X,x,A,y,B), G 2 =G(X,x,A∙B,y,b) → Theorem: For G 1 =G(X,x,A,y,B), such that P X-x =P X-y, sgn(λ 2 (G 1 )-r)= =sgn(λ 2 (β(G 1 ))-r) holds.

ω - transformation Let T be the family of connected graphs G=C(X,A,B) with at least one triangle, whose subgraphs X, A and B are rooted graphs, with the roots x, a and b, respectively, where λ 1 (X)=r (r > 0). ω: T →V, ω(G 1 )=G 2 where G 1 =C(X,A,B), G 2 =G(X,A,B) → Theorem: For G 1 =C(X,A,B), if sgn(λ 2 (ω(G 1 ))-r) is equal to 0 or -1, then sgn(λ 2 (G 1 )-r) is equal to -1.

τ - transformation Let T’ be the subfamily of the family T containing graphs G=C(X,A∙X’,B) such that the rooted subgraph X’ has the property λ 1 (X’)=r (r > 0). τ: T’ →V, τ(G 1 )=G 2 where G 1 =C(X,A∙X’,B), G 2 =G(X,A,B) → Theorem: For G 1 =C(X,A∙X’,B), sgn(λ 2 (G 1 )-r)=sgn(λ 2 (α(G 1 ))-r) holds.

φ - transformation Let Q be the family of connected graphs G=C(X,A,B,X’) with at least one quadrangle, whose subgraphs X, A, X’ and B are rooted graphs, with the roots x, a, x’ and b, respectively, where λ 1 (X)= λ 1 (X’)= r (r > 0). φ: V →V, φ(G 1 )=G 2 where G 1 =C(X,A,B,X’), G 2 =G(X,A,B) → Theorem: For G=C(X,A,B,X’), P G (r)=0 holds. Also, the following equivalences hold: 1) 2)

Applications – one example Four families of maximal reflexive cacti in the class of bicyclic graphs with the bridge: A 1 -A 14, with more than one vertex different from c-vertices additionally loaded B 1 -B 11, with exactly one white vertex different from c-vertices additionally loaded C 1 -C 41, with exactly one black vertex different from c-vertices additionally loaded D 1 -D 36, with only c-vertices additionally loaded

Smith graphs

D 1 -D 35 α – transformations: Exception:

B 1 -B 11 and C 1 -C 41 α and β – transformations:

Thank you!