Laplacian Spectral Properties of Signed Circular Caterpillars

A circular caterpillar of girth n is a graph such that the removal of all pendant vertices yields a cycle Cn of order n. A signed graph is a pair Γ = (G,σ), where G is a simple graph and σ ∶ E(G) → {+1,−1} is the sign function defined on the set E(G) of edges of G. The signed graph Γ is said to be balanced if the number of negatively signed edges in each cycle is even, and it is said to be unbalanced otherwise. We determine some bounds for the first n Laplacian eigenvalues of any signed circular caterpillar. As an application, we prove that each signed spiked triangle (G(3;p, q, r), σ), i. e. a signed circular caterpillar of girth 3 and degree sequence πp,q,r = (p + 2, q + 2, r + 2,1, . . . ,1), is determined by its Laplacian spectrum up to switching isomorphism. Moreover, in the set of signed spiked triangles of order N , we identify the extremal graphs with respect to the Laplacian spectral radius and the first two Zagreb indices. It turns out that the unbalanced spiked triangle with degree sequence πN−3,0,0 and the balanced spike triangle (G(3; p̂, q̂, r̂),+), where each pair in {p̂, q̂, r̂} differs at most by 1, respectively maximizes and minimizes the Laplacian spectral radius and both the Zagreb indices.


Introduction
A signed graph Γ is a pair (G, σ), where G = (V (G), E(G)) is a graph and σ ∶ E(G) → {+1, −1} is a sign function (or signature) on the edges of G. The (unsigned) graph G of Γ = (G, σ) is called the underlying graph. Each cycle C in Γ has a sign given by σ(C) = ∏ e∈C σ(e). A cycle whose sign is 1 (resp. −1) is called positive (resp. negative). A signed graph (G, σ) (and its signature σ as well) is said to be unbalanced if it contains at least one negative cycle, and balanced otherwise. In particular, the null graph K 0 with one vertex and 0 edges is a balanced signed graph. The signed graph obtained from Γ by switching signs to all its edges is denoted by −Γ. If all edges in Γ are positive, we write Γ = (G, +), and set (G, −) = −(G, +).
The reader is referred to [14] for basic results on the graph spectra and to [21] for basic results on the spectra of signed graphs.
Many familiar notions related to unsigned graphs directly extend to signed graphs. For example, Γ = (G, σ) is said to be k-cyclic if the underlying graph G is k-cyclic. This means that G is connected and its cyclomatic number E(G) − V (G) + 1 is equal to k. The words unicyclic and bicyclic stand as synonyms for 1-cyclic and 2-cyclic respectively. Moreover, if G is neither a tree or a forest, the girth (resp. circumference) of Γ is the length of the shortest (resp. longest) cycle contained in G.
The degree sequence (d 1 , . . . , d N ) of Γ is simply the non-increasing sequence of vertex degrees in G. We recall that a vertex v is said to be pendant (resp. internal) if its vertex degree d G (v) equals 1 (resp. d G (v) > 1). A quasi-pendant vertex is instead a vertex adjacent to a pendant vertex.
The signed adjacency matrix A(Γ) is obtained from the usual adjacency matrix of the underlying graph G by replacing 1 with −1 whenever the corresponding edge is negative.

Preliminaries
We start by recalling a characterization for switching equivalent signatures. Let Γ = (G, σ) be a signed graph of order N . The Laplacian eigenvalues, i.e. the roots of the Laplacian polynomial ψ(Γ, x) = det(xI − L(Γ)), are all real since L(Γ) is symmetric and are denoted by The last inequality holds since the Laplacian matrix is positive semidefinite. The following two results are surely known to the experts. We provide a proof for both of them for sake of completeness.
Proposition 2.2. Let Γ = (G, σ) be any signed graph of order N , and let Γ ′ = (H, σ H ) be any subgraph of Γ. The following inequalities hold: Proof. Let Γ − E denote the signed graph obtained from Γ by deleting the edges in the set E ⊆ E(G). From the ordinary vertex variant interlacing theorem for the adjacency matrix combined with [10, Theorem 2.3(ii)] we deduce that, for every {e} ⊆ E(G), the L-eigenvalues of Γ and those of Γ − {e} interlace as follows: Inequalities 2 are stated, for instance, in [10,Theorem 2.5]. Note now that Γ ′ shares the same non-zero eigenvalues of a suitable graph of type Γ − E. Hence, (1) comes from (2) used E(G) − E(H) times.
Corollary 2.3. Let ∆ be the largest vertex degree of a signed graph Γ = (G, σ). Then, Proof. The star K 1,∆ is a subgraph of G. Since K 1,∆ is a tree, Proposition 2.1 implies that all signatures defined on it are switching equivalent. Hence, we get The statement now follows from Proposition 2.2.
In [9] Belardo and Simić contrived a geometric-combinatorial way to compute the several coefficients of ψ(Γ, x). In order to describe such achievement, we need to recall that a TUsubgraph of any fixed signed graph Γ is a subgraph whose components are trees or unbalanced unicyclic graphs. In other words, a TU-subgraph H admits a vertex disjoint decomposition where, if any, the T i 's are trees and the U j 's are unbalanced unicyclic graphs. The weight of the signed TU-subgraph H is defined as Theorem 2.4. [9, Theorem 3.9] Let ψ(Γ, x) = x N +b 1 x N −1 +⋯+b N −1 x+b N be the L-polynomial of a signed graph Γ. Then, the equality where H i denotes the set of signed TU-subgraphs of Γ containing i edges, holds for all i = 1, 2, . . . , N .
Let {v 1 , . . . , v N } the vertex set of a signed graph Γ = (G, σ), and let t + Γ and t − Γ respectively denote the number of balanced triangles and unbalanced triangles contained in Γ. We set Theorem 2.5 reveals to be very helpful to detect possible L-cospectral mates of Γ. (ii) Γ and Λ have the same number of balanced components; (iii) Γ and Λ have the same Laplacian spectral moments T k = ∑ N i=1 µ k i , for all non-negative integers k; (iv) Γ and Λ have the same sum of squares of degrees, i.e. f 1 (Γ) = f 1 (Λ); (v) f 2 (Γ) = f 2 (Λ).
Given any signed cycle (C, σ), we set In the statement of Proposition 2.6 below, u ∼ v means that two vertices u and v in a signed graph Γ are adjacent; uv denotes the edge connecting them; C v is the set of cycles in G passing through a fixed vertex v; and C uv is the set of cycles in G containing uv among their edges. Proposition 2.6. Let Γ = (G, σ) be any signed graph. The following equations hold: Proof. As explained at the end of Section 2 in [10], the Laplacian matrix of a signed graph Γ = (G, σ) can be regarded as the adjacency matrix of a weighted multigraph Γ * sharing with Γ the vertex set V (G). The positive (resp. negative) edges of Γ correspond to the (−1)-weighted (resp. (+1)-weighted) edges of Γ * . Moreover, if a vertex v of Γ has degree k, the graph Γ * has a k-weighted loop at v. Formulae 5 and 6 now come from [10, Theorem 2.9] applied to Γ * .
We end this section of preliminaries by recalling a very well-known result on the determinant of a 2 × 2 block matrix.

Circular caterpillars
As recalled in Section 1, a graph G is said to be a circular caterpillar if its internal vertices induce a cycle. We say that a circular caterpillar is complete if each internal vertex is quasipendant. It is immediately seen that a circular caterpillar is complete if and only if no vertices have degree 2. Letσ be any fixed unbalanced signature on a circular caterpillar G. As a consequence of Proposition 2.1, every circular caterpillar, being unicyclic (and, hence, containing just one cycle), admits only two different non-equivalent signatures. In other words, all unbalanced signatures are equivalent toσ. It not restrictive to assume that (G,σ) has just one negative edge. Such edge necessarily connects two internal vertices. Our first result concern the bidegreed circular caterpillar U (∆, n). This is the only circular caterpillar of girth n whose internal vertices have all degree ∆. The graph U (∆, n) has n(∆ − 1) vertices. We are borrowing the notation from [3,5,6], where extremality of the adjacency spectral radius of U (∆, n) with respect to a suitable class of (unsigned) graphs is discussed. Graphs in the set {U (3, n) ∀ n ⩾ 3 } are also known as sun graphs.
To lighten notation we shall denote by C σ n the subgraph of (U (∆, n), σ) induced by its internal vertices.
Theorem 3.1. Let ∆ ⩾ 3. The Laplacian spectrum of (U (∆, n), σ) contains 1 with multiplicity n(∆ − 3). The remaining 2n L-eigenvalues are given by Proof. Up to possibly replacing σ with a switching equivalent signature, we can assume that all edges not belonging to the cycle C n are positive. Let k = ∆ − 2. We label the vertices of (U (∆, n), σ) assigning the highest labels to the internal vertices. Thus, vertices v nk+1 , . . . , v nk+n belong to the subgraph C n . Moreover, we suppose that, for each h ∈ {1, ..., n}, the k vertices v (h−1)k+1 , . . . , v hk of degree 1 are adjacent to v nk+h (see Fig. 1). Let 1 k and I c be the all 1's column vector of size k and the (c × c)-identity matrix respectively. The Laplacian matrix for U (∆, n) takes the following form: Since (P nk×n ) T P nk×n = kI n , Proposition 2.7 applied to This is the reason why every root of is an L-eigenvalue of (U (∆, n), σ). So far, we have proved that the L-spectrum of (U (∆, n), σ) contains the roots of the polynomials It is now elementary to check that such roots are given by (7).
As an immediate consequence of Theorem 3.1, the absence of 1 in the Laplacian spectrum characterizes the sun graphs among all bidegreed circular caterpillars. The following two results help to locate on the real line the several elements of Spec L (U (∆, n), σ).
n be a signed cycle.
i) The Laplacian spectral radius is given by: In the latter case, the Laplacian spectral radius has multiplicity 2.
We now prove Part ii). The Descartes' rule of signs confirms that the two roots φ − (λ, ∆) and where the equality holds if and only if λ = 0. Such value belongs to Spec L (C σ n ) if and only if the circular caterpillar is balanced. The proof ends once you note that, by Equation (10), Let Γ be any signed circular caterpillar in CC n not equal to a signed cycle. We denote by ∆ Γ its maximum vertex degree, and by δ Γ its least vertex degree bigger than 2. Next theorem gives bounds for the first n Laplacian eigenvalues of Γ.
In particular, Moreover, if Γ is complete, then Proof. In our hypotheses G is a subgraph of U (∆ Γ , n) and 3 ⩽ δ Γ ⩽ ∆ Γ . If Γ is additionally complete, U (δ Γ , n) is a subgraph of G. Whichever signature we choose on U (∆ Γ , n) to extend σ, balancedness (resp. unbalancedness) is preserved. The inequalities of the statement now come from Proposition 2.2, Theorem 3.3, and the fact that all unbalanced signatures on unicyclic graphs are equivalent.
Let (G, σ) be a signed graph, and let G be a subgraph of H. The more natural way to extend σ to H is to define For any signed graph Γ = (G, σ), we denote by θ(Γ) the number of its Laplacian eigenvalues bigger that 1.

Proof. By Theorems 3.3 and 3.4 we get
If, additionally Γ is complete, then For the rest of this section we collect some more results giving further constraints on a signed graph Λ in order to be L-cospectral to a circular caterpillar.
Proposition 3.6. Let Γ = (G, σ) be a circular caterpillar of order N and girth n. The last two coefficients of the Laplacian polynomial ψ(Γ, x) can be read on Table 1.
Proof. Note that E(G) = N , G being unicyclic. According to Theorem 2.4, we get where H k is the set of T U -subgraphs with k edges contained in Γ. Now, and γ(Γ) = 4 by (3). This justifies the last column of Table 1. We are left to investigate the number and the geometric nature of elements in H N −1 . When Γ is balanced, it does not contain any unbalanced unicyclic subgraph, Hence H N −1 just contains the pairwise distinct trees in {T i 1 ⩽ i ⩽ n} of order N which are obtained by removing exactly one edge of the cycle C n inside G. Thus, by (3), If instead Γ is unbalanced, in addition to the trees T i 's, H N −1 also contains the unbalanced unicyclic graphs U j 's obtained from Γ by removing an edge not belonging to C n . therefore, we find N − n pairwise different U j 's, and, by (3), γ(U j ) = 4 for 1 ⩽ j ⩽ N − n. Hence, as claimed.
Corollary 3.7. A signed graph Λ which is L-cospectral to an unbalanced circular caterpillar is necessarily connected.
Proof. Let c be the number of connected components of Λ. By Theorem 2.5 (ii) each component of Λ is unbalanced. It follows that Λ has no trees among its components. Together with Theorem 2.5 (i), this implies that each component of Λ is also unicyclic. We now use Theorem 2.4 and Proposition 3.6 to get hence, c = 1 as claimed.
Lemma 3.8. Let P s be the path with s vertices. Whatever signature σ we choose on P s , we have θ(P s , σ) = ⌈ 2s 3 ⌉ − 1.
Proposition 3.9. Let Λ = (H, τ ) be a graph which is L-cospectral to a graph Γ in CC n . Then, Proof. Let d = diam(Λ). By definition, H contains the path P d has a subgraph. Thus, by Lemma 3.8 and Corollary 3.5, By analyzing the several mod 3 cases for d, we get the statement.
We now focus our attention on elements in CC 3 , i.e. on signed spiked triangles. For every N ⩾ 3, we denote by CC 3 (N ) the set of signed spiked triangles of order N , and by G(3; p, q, r) the unique circular caterpillars of girth 3 and degree sequence π p,q,r = (p+2, q+2, r+2, 1, . . . , 1). Obviously, the graph G((3; p, q, r), σ) belongs to CC 3 (p+q+r+3). As noted at the beginning of Section 3, all unbalanced signatures on G(3; p, q, r) give rise to the same Laplacian spectrum.
Proof. Let P (k) ∶= x 2 − (k + 3)x + 2. The equality (11) can be reached using Proposition 2.6 through the following steps: employ first (5) for v being one of the three vertices of the cycle C 3 in G(3; p, q, r). Afterwards, use either (5) or (6) on the resulting summands yet to expand. It turns out that ψ(Γ, x) is equal to The statement now comes by the definition of ω (see (4) above) and P (k). Proof. In the light of the remarks made in Section 1, we just need to prove the 'only if' part. Let Γ = (G(3; p, q, r), σ) and Γ ′ = (G(3; p ′ , q ′ , r ′ ), σ ′ ) be two L-cospectral circular caterpillars.
The presence or the absence of 0 in their common Laplacian spectrum allows to establish whether Γ and Γ ′ are both balanced or both unbalanced. LetÑ 3 0 be the set of non-increasing triples of non-negative integers. The coefficients of ψ + (p, q, r)(x) and ψ − (p, q, r)(x) are peculiar linear combinations of the four elementary symmetric polynomials evaluated at (p, q, r) ∈Ñ 3 0 . The two polynomial ψ(Γ, x) and ψ(Γ ′ , x) are equal only if Such equalities actually occur only if the non-increasing sequences (p, q, r) and (p ′ , q ′ , r ′ ) are equal. In order to see this, observe that the map is injective. In fact, Θ maps different non-increasing triples to cubic polynomials with different zero sets. The non-negative numbers s i (a, b, c) for i = 1, . . . , 4 are precisely the moduli of the coefficients in Θ(a, b, c). Proof. The statement follows from a direct comparison between the Laplacian spectra of the relatively small list of signed graphs of order N ⩽ 6. Proof. Since N ⩾ 7, the number p is at least 2. It follows that G(3; p, q, r) contains K = G(3; 2, 0, 0) as subgraph. Recalling Proposition 2.2, where the last equality is due to Theorem 3.3 ii).
Recall that, given any signed graph Γ, the number θ(Γ) counts the number of eigenvalues bigger than 1 in Spec L (Γ). Throughout the rest of the paper, Ω k will denote the set of connected signed graphs Γ such that θ(Γ) = k, and U is understood to be the set of all signed graphs which are L-cospectral to some signed spiked triangle. Proof. Let Γ ∈ CC 3 be a spiked triangle which is L-cospectral to Λ. By Proposition 3.9, it follows that diam(Λ) ⩽ 3 2 ⋅ 4 = 6, whereas Corollary 3.5 yields θ(Λ) ⩽ 3.
The following lemma, though known to the experts, has been inserted with a proof for sake of clarity.
In order to achieve Proposition 4.9 ensuring the connectivity of all signed graphs in U, the following intermediate lemma will be helpful.
Lemma 4.7. Let Λ 1 , . . . , Λ h be the connected components of Λ = (H, τ ) ∈ U ordered in a nondecreasing fashion with respect to the cyclomatic number. If the Λ i 's are not all unicyclic, then h > 1; Λ 1 is a tree; Λ h is bicyclic and unbalanced; and, when h > 2, Λ i is unicyclic and unbalanced for 2 ⩽ i ⩽ h − 1.
Proof. From Theorem 2.5 (ii), we know that V (H) = E(H) . Because of such equality, for each k-cyclic signed graph with k > 1 in the set Υ = {Λ i 1 ⩽ i ⩽ h}, we also find in it k − 1 trees. By Theorem 2.5 (ii), the set Υ contains at most one balanced component, and a fortiori at most one tree. Hence, k − 1 ⩽ 1, and Λ has at most one bicyclic component. Proof. Lemma 4.7 guarantees that no signed graphs in U have k-cyclic components with k ⩾ 3. Moreover, by Theorem 2.5 (i), we know that V (H) = E(H) for all Λ = (H, τ ) ∈ U. This implies in particular that U does not contains trees or forests.
The rest of the proof is devoted to show that no signed graphs with circumference c > 3 lie in U.
Consider the set S of all signed graphs switching isomorphic to either (C 6 , +); (C 5 , −); (C 4 , +); (C 4 , −) or one of the graphs in Fig. 2. If Γ is a k-cyclic signed graph with k ∈ {1, 2}, circumference c ⩾ 4 and θ(Γ) ⩽ 3, then Γ belongs to S. This fact relies upon Proposition 2.2, once you check that: • θ(C 5 , +) = θ(C 6 , −) = 4, θ(C n , σ) ⩾ 4 for all n ⩾ 7 (see Proposition 3.2); • for every 1-cyclic or 2-cyclic signed graph Γ ′ obtained by adding an additional edge (and possibly an additional vertex) to a graph Γ in S, we have θ(Γ ′ ) = 4 unless Γ ′ ∈ S. By Theorem 2.5 (i) and Proposition 4.3, the intersection U ∩ S is empty. We now claim that there are no non-connected graphs in U having elements in S among their components. Assume by contradiction that there exists a non-connected Λ ∈ U such that one of its components, say Λ ′ , is in S, and consider the set of signed graphs T obtained from S by subtracting the switching isomorphic copies of (C 4 , −). Note that the function θ is additive with respect to the union of disjoint graphs, and θ(Γ) = 3 for all graphs in T . By Theorem 2.5 (ii), Lemma 4.5 and Lemma 4.6, we infer that if Λ ′ ∈ T , then Λ = Λ ′ ∪K 0 and Λ ′ is unbalanced. It follows that Λ would have at most 6 vertices, but signed spiked triangles with at most 6 vertices are DLS; hence, up to switching equivalence, Λ ′ = (C 4 , −); thus, θ(Λ ′ ) = 2.
Along the proofs of Proposition 4.9 and Theorem 4.10 we denote by L 3,N the lollipop graph with girth g and order N , i.e. the unsigned graph obtained by attaching a pendant path P N −2 to a vertex of the triangle C 3 . Proposition 4.9. Every signed graph in U is unicyclic.
Proof. By Theorem 2.5 (i), it suffices to show that all signed graphs in U are connected. Connectedness is already ensured for items in U which are either L-cospectral to an unbalanced spiked triangle (see Corollary 3.7) or have at most 6 vertices (see Proposition 4.3).
Assume now that Λ = (H, τ ) ∈ U has at least 7 vertices and is switching isomorphic to a balanced spiked triangle. Thank to Theorem 2.5 (ii) and Proposition 4.8 respectively, we know that Λ has just one balanced component and circumference c = 3. Moreover, θ(Λ) ⩽ 3 by Lemma 4.5.
We first prove that Λ cannot have a bicyclic component. Suppose the contrary. Fig. 3 describes all bicyclic graphs Γ of circumference 3 such that θ(Λ) ⩽ 3, where it is understood that N ⩾ 5 is the order of the graphs Γ 12 (N ), Γ 13 (N ) and Γ 14 (N ), all having N − 5 pendant vertices.
When evaluated in 0 and in 3, the polynomials both give negative values. A straightforward calculus argument shows that µ 2 (Γ 13 (N )) = µ 2 (Γ 14 (N )) = 3. Since We check directly that no pairs in the set of signed graphs are L-cospectral, and, for N ⩾ 6, we already observed that Γ 13 (N ) and Γ 14 (N ) have at least one eigenvalue belonging to the interval (0, 1). We conclude that there is not a Λ ∈ U with a bicyclic connected component. So far, we have proved that if Λ ∈ U is not connected, just one among its connected components Λ 1 , . . . , Λ h is balanced, and none of them is bicyclic. By Lemmas 4.7 and 4.6 it follows that each Λ i is unicyclic and has circumference c = 3.
Suppose that Γ(s, t) = (G(s, t), σ) is unbalanced, and let Γ be any unbalanced spiked triangle of order N . Theorem 2.4 and Proposition 3.6 give This proves that all unbalanced spiked triangles are DLS.
Consider now the balanced graph Γ(s, t) = (G(s, t), +). The multiset Spec L (Γ(s, t)) contains 3 for all s ⩾ 3 and t ⩾ 2. This fact is due to the presence in Γ(s, t) of a 'pendant triangle'. Through MATLAB or a manual polynomial long division, we discover that if 3 belongs to Spec L (G(3; p, q, r), +), then 2(pq + pr + qr) = 3pqr.
The proof will be over, once we show that the signed graphs in (18) are not L-cospectral to a graph of type Γ(s, t). To this aim we consider, for every signed graph Γ = (G, σ), the triple of non negative integers where the functions f 1 and f 2 have been defined just before the statement of Theorem 2.5. By Parts (i), (iv) and (v) of Theorem 2.5, the function Ψ should return the same triples when evaluated on pairs of L-cospectral graphs, and elementary algebraic manipulations show that no integers s and t exist such that Ψ(Γ(s, t)) = (s + t, s 2 + t 2 + s + t + 4, s 3 + t 3 + s + t + 6) is equal to one of the triples

Extremal spiked triangles
Now that the Laplacian spectral characterization of signed spiked triangles is over, we solve the problem of finding extremal elements in CC 3 (N ) with respect to certain topological indices.
Proof. Let G be a connected graph of order N = p + q + r + 3 and degree sequence π p,q,r = (p + 2, q + 2, r + 2, 1, . . . , 1). The graph G is necessarily unicyclic; in fact the sum of vertex degrees gives 2 E(G) . Therefore, Since there are only 3 vertices whose degree is bigger than 1, the girth of G is necessarily 3, and G = G(3; p, q, r).
Proof. Unicity of (p 1 , . . . ,p n ) comes from the Division algorithm: since there exists a unique pair (s, r) ∈ N 2 0 such that k = ns + r and 0 ⩽ r < n, we necessarily havê When either n ⩽ 2 or k ⩽ 2, there is nothing else to prove, sinceÑ 1 0 (k) is a singleton, ◃ is a total ordering onÑ 2 0 (k) for all k > 0, andS n (1) andS n (2) are empty for all n > 0. Let now n > 2 and k ⩾ 3. The second majorization is immediate. In order to prove the first one, we assume by contradiction that, for some fixed k, the set {(p 1 , . . . , p n ) ∈S n (k) (p 1 , . . . ,p n ) ◃ (p 1 , . . . , p n )} is not empty and denote by (q 1 , . . . , q n ) its minimum with respect to the lexicographic order. Let us distinguish two cases.