Least eigenvalue of the connected graphs whose complements are cacti

Abstract Suppose that Γ is a graph of order n and A(Γ) = [ai,j] is its adjacency matrix such that ai,j is equal to 1 if vi is adjacent to vj and ai,j is zero otherwise, where 1 ≤ i, j ≤ n. In a family of graphs, a graph is called minimizing if the least eigenvalue of its adjacency matrix is minimum in the set of the least eigenvalues of all the graphs. Petrović et al. [On the least eigenvalue of cacti, Linear Algebra Appl., 2011, 435, 2357-2364] characterized a minimizing graph in the family of all cacti such that the complement of this minimizing graph is disconnected. In this paper, we characterize the minimizing graphs G ∈ Ωnc $\begin{array}{} {\it\Omega}^c_n \end{array}$, i.e. λmin(G)≤λmin(Cc) $$\begin{array}{} \displaystyle \lambda_{min}(G)\leq\lambda_{min}(C^c) \end{array}$$ for each Cc ∈ Ωnc $\begin{array}{} {\it\Omega}^c_n \end{array}$, where Ωnc $\begin{array}{} {\it\Omega}^c_n \end{array}$ is a collection of connected graphs such that the complement of each graph of order n is a cactus with the condition that either its each block is only an edge or it has at least one block which is an edge and at least one block which is a cycle.


Introduction
Let Γ = (V(Γ), E(Γ)) be a graph such that V(Γ) = {v i : ≤ i ≤ n} and E(Γ) are set of vertices and edges respectively. Assume that all the considered graphs are simple, nite and undirected. For each i, the degree d(i) is the number of incident edges on v i . The adjacency matrix of Γ is A(Γ) = [a i,j ] with a i,j equal to if v i is linked to v j and a i,j is zero for the rest case, where ≤ i, j ≤ n. The solutions of det(A(Γ) − λI) = are eigenvalues of Γ. It is interesting to note that A(Γ) is always symmetric and real, all the eigenvalues can be arrange as λ (Γ) ≤ λ (Γ) ≤ ... ≤ λn(Γ). The eigenvectors corresponding to the least eigenvalue λ (Γ) and the greatest eigenvalue (spectral radius) λn(Γ) are called rst eigenvector (FEv) and Perron-vector respectively.
The spectrum of the adjacency matrix for an undirected graph is rst time studied by Collatz and Sinogowitz (1957), see [1]. Later on, many researchers discussed the largest eigenvalue (spectral radius) in the area of spectra of graphs, see [2,3]. It is observed that the least eigenvalue did not receive the attention of researchers as compare of the largest eigenvalue. From the few of results of the least eigenvalues on the graphs, the bounds related results can be found in [4,5]. For further study, we refer [6][7][8][9][10][11][12][13]. A graph Γ is said to be minimizing in a certain collection of graphs, if the least eigenvalue of A(Γ) is minimum in the set of all the least eigenvalues of the other graphs in the same collection.
Let G(p, q) be a collection of connected graphs in which each graph is of p order and q size such that < q < p(p− ) . The minimizing graph in G(p, q) characterized by Bell et al. [14] is stated in the below result: Theorem 1.1. Minimizing graph in G(p, q) is either a join of two nested split graphs, or a bipartite graph.
It is important to note that the complements of the graphs characterized by Bell contain the cliques such that order of each clique is greater or equal to p or these are disconnected. After it, the question is raised to investigate the minimizing graphs in the collection of connected graphs such that the complement of each graph contains the cliques of small sizes. Motivated by it, the minimizing graph in the collection of connected graphs such that the complement of each graph is trees, unicyclic or bicyclic are characterized by Fan, Zhang, Wang, Li and Javaid, see [15][16][17][18]. For further study, we refer to [19 -22]. In this paper, the minimizing graph is characterized in the collection of connected graphs such that the complement of each graph is cactus with the condition that each block of a cactus of order n is only an edge or a cactus of order n has at least one block which is an edge and at least one block which is a cycle. In the rest of paper; Section 2 includes some basic de nitions and terminologies, Section 3 contains the proofs of some important lemmas and Section 4 has the main results in which minimizing graph is characterized in the family of connected graphs with the condition that the complement of each graph is cactus.

Preliminaries
A connected graph is called cactus if and only if its every block is either a simple cycle or a single edge. A cactus is a tree if and only if its each block is an edge. An edge of a cactus is a cycle edge if it is in some cycle, and tree edge, otherwise. A cactus is said to be a bundle if there is a single common vertex on all of its cycles. Let B (n) be the bundle of order n + obtained from a star K ,n of the same order with central vertex V by adding the edges v i v i+ , where i ∈ { , , , ..., n − } and n = mod( ). Thus, the central vertex v of the bundle B (n) has degree n and each remaining vertex is of degree . Similarly, let B (n) be a bundle obtained from K ,n by adding the edges v i v i+ , where i ∈ { , , , ..., n − }, n = mod( ) and |V(B (n))| = n + . Thus, for the vertices of B (n), d(v ) = n, d(vn) = and d(v i ) = , where ≤ i ≤ n − .
We de ne some particular cacti which are obtained from the aforesaid bundles.
De nition 2.1. Assume that p, q = mod( ) are positive integers. Let B (p) and B (q) be two bundles. The cactus graph C (p, q) is constructed by the join of a vertex of B (p) with a vertex B (q), where both the vertices are of degree . Thus, Assume that p, q ≡ mod( ) and p, q ≥ . If a vertex of the bundle B (p) is joined with a vertex of the bundle B (q) then we obtain the cactus graph C (p, q), where both the chosen vertices are pendent and n = + p + q = |C (p, q)|. For p ≥ , q ≥ , p ≡ mod( ) and q ≡ mod( ), if we join a vertex of the bundle B (p) to a vertex of the bundle B (q) then we obtain a cactus graph C (p, q), where the chosen vertices are of degree 1 and 2 respectively and n = + p + q = |C (p, q)|. Similarly, if we assume p ≥ , q ≥ , p = mod( ) and q = mod( ), and choose two vertices of degree and in B (p) and B (q) respectively. On joining these chosen vertices by an edge, we obtain the cactus graph C (p, q) with n = + p + q = |C (p, q)|.
We note that C (p, q) ∼ = C (q, p) and C (p, q) ∼ = C (q, p) as p and q both are even in C (p, q) and odd in C (p, q). Moreover, as p is odd and q is even in C (p, q), and p is even and q is odd in C (p, q) therefore C (p, q) ̸ ∼ = C (q, p), C (p, q) ̸ ∼ = C (q, p) and C (q, p) ∼ = C (p, q). The cacti C (p, q) and C (p, q) are presented in Figure 1((a) and (b)) and the cacti C (p, q) and C (p, q) are presented in Figure 2((a) and (b)).
Let Ω ,n be the class of cacti other than stars such that each block of a cactus is an edge and Ω ,n be a class of cacti other than bundles such that at least one block of each cactus is an edge and at least one  block is a cycle. Let Ωn be a class of cacti other than stars and bundles such that either all the blocks of a cactus are edges or a cactus has at least one block which is a cycle and at least one block which is an edge, i.e. Ωn = Ω ,n ∪ Ω ,n . Thus, we obatain Ω c n = {Γ c : Γ c is connected, |Γ c | = n ∧ Γ ∈ Ωn}. By interlacing theorem, λ min (Γ) ≤ − if Γ contains at least one edge. Moreover, equality holds if Γ is a complete graph. Another way to achieve this equality is if Γ = ∪ i G i , where all G i are complete graphs and at least one G i is non-trivial. Thus, for Γ ∈ Ωn, λ min (Γ c ) < − .
If ϕ : V(Γ) → {X i : ≤ i ≤ n} is a 1-1 map such that ϕ (u i ) = X i for each u i ∈ V(Γ) then it is said to be de ned on the graph Γ. The eigenvector X of A(Γ) is naturally de ned on V(Γ). Thus, we have where all adjacent to v are in N Γ (v). If X ∈ R n is an arbitrary unit vector, we have where equality holds i X is a FEv. If Γ c is complement of Γ, then A(Γ c ) = J − I − A(Γ) with J and I as all-ones and identity matrix respectively. Thus, for X ∈ R n with least root λ .

Minimizing graphs
Now, we present some important lemmas of the minimizing graph which are frequently used in next section. The classes of cacti which have graphs of even order are discussed from Lemma 3.1 to Lemma 3.6. Moreover, the cacti of odd order are studied from Lemma 3.7 to Lemma 3.10. Firstly, we discuss the classes of cacti which have graphs of even order.
Lemma 3.1. Suppose that p, q ≥ , n ≥ are integers with p, q, n ≡ (mod ). If p > q + , then As p is greater than q + and λ is less Proof. Since, C (p, q) c ∼ = C (q, p) c , therefore proof is same as of Lemma 3.1.

Lemma 3.3.
Suppose that p, q ≥ are integers with p, q ≡ (mod ) and p + q where equality holds i p = n− = q with n ≥ and p = n and q = n− with n ≥ .
where p + q + = n is cardinality of both the cacti.
Consequently, for n ≥ and n ≡ (mod ), we have λ min (C ( n , n− ) c ) ≤ λ min (C (p, q) c ) with equality i p = n and q = n− , and (b) for n ≥ and n ≡ (mod ), we have λ min (C ( n− , n− ) c ) ≤ λ min (C (p, q) c ) with equality i p = n− = q. Now, we discuss the classes of graphs having graphs of odd order.
Lemma 3.10. Suppose that p ≥ and q ≥ are integers with p ≡ (mod ) and q ≡ (mod ), where p + q + = n is of the cacti. Then where equality holds i p = n+ and q = n− with n ≥ , and p = n− and q = n− with n ≥ .
Consequently, for n ≥ and n ≡ (mod ), we have λ min (C ( n+ , n− ) c ) ≤ λ min (C (p, q) c ) with equality if p = n+ and p = n− , and (b) for n ≥ and n ≡ (mod ), we have λ min (C ( n− , n− ) c ) ≤ λ min (C (p, q) c ) with equality i p = n− and q = n− . This complete the proof.

Characterization
This section includes the main results in which minimizing graphs are characterized in the family of connected graphs with the condition that the complement of each graph is a cactus such that either its each block is only an edge or it has at least one block which is an edge and at least one block which is a cycle. In

Proof.
Suppose that X is non-negative and discuss the following two cases: Case-1. Suppose that C ∈ Ωn is a cactus graph such that its each block is an edge i.e. C ∈ Ω ,n . Let X be the value of the vertex v ∈ C assigned by X. As n ≥ and is not a bundle, therefore there exist a vertex in C say u which is not adjacent to v. Thus, a vertex w adjacent to u (w ∼ u) exists on a path from v to u as C is connected. A new cactusC having each block as an edge can be found on the deletion of the edge wu and addition of vu in C. We nd a star K ,n− with center v by repeating the same process for the non-neighbor of v in the cactusC and so on. Thus, we have Case-2. Suppose that C ∈ Ωn is a cactus graph such that it has at least one block which is an edge and at least one block which is a cycle i.e. C ∈ Ω ,n . Assume that k is number of cycles in C, then ≤ k ≤ n− , where n is order of C. Now, we delete an edge in each cycle under the conditions, (i) no two deleted edges of any two cycles have common vertex and (ii) the deleted edge in a cycle which has the vertex with label X is not incident on this vertex. Thus, we have a cactus graph such that its each block is an edge, say C i.e. C ∈ Ω ,n . Then by Case 1, we have Xu Xv . Now, we add the deleted edges, then Xu Xv + (Xv r Xv r + + Xv r Xv r + +, ..., +Xv r k Xv r k + ) Xu Xv + (Xv r Xv r + + Xv r Xv r + +, ..., +Xv r k Xv r k + ), where vr , v r+ , vs , v s+ , ..., v t , v t+ are k distinct vertices of C. Moreover, the inequality in (4.1) does not disturb, if we add k terms (Xv r Xv r + + Xv r Xv r + +, ..., +Xv r k Xv r k + ) in its right hand side. Consequently, from both the cases, we have Xu Xv + (Xv r Xv r + + Xv r Xv r + +, ..., +Xv r k Xv r k + ).

Lemma 4.2. For n ≥
and C c ∈ Ω c n , the rst eigenvector X of C c has at least negative and positive entries.

Proof.
Assume that there is a unique vertex v ∈ C c having positive value labeled by X. The degree of v in C c is non-zero, i.e d C c (v) ≠ . As, if d C c (v) = , then C is a bundle, which is a contradiction to the construction of Ω c n . Consequently, ≤ d C c (v) ≤ n − . Let u be another vertex in C c . Since all the vertices are supposed to have negative values except v, we claim u ∼ v, otherwise (2.2) does not holds for u ∈ C c as λXu > and It shows that C is disconnected, which is a contradiction to the construction of Ω c n . Similarly, we can prove, if v ∈ C c is a unique vertex with negative value assigned by X. (a) Assume that n ≡ (mod ), where n = p + q + . LetC be a graph obtained from C by some possible addition or deletion of edges in C+ and C− such that the subgraphC+ andC− ofC induced by C+ and C− are cactus graphs satisfying one of the following possibilities, (i) each block of one of the subgraphsC+ andC− is an edge and other has at least one block which is a cycle and at least one block which is an edge, (ii) all the blocks of both the subgraphsC+ andC− are cycles, (iii) all the blocks of one subgraph are edges and of other are cycles, (iv) each block of one of the subgraphsC+ andC− is a cycle and other has at least one block which is a cycle and at least one block which is an edge, and (v) both the subgraphsC+ andC− have at least one block which is a cycle and at least one block which is an edge. For (i), supposeC+ is a cactus such that its each block is an edge, otherwise we take −X as a rst eigenvector. Let u be a vertex ofC+ with maximum modulus among all the vertices, then by discussion of equation (4.1) in Lemma 4.1, we obtain a cactus with each block as an edge which is infect a star K ,p . Similarly, suppose that v is a vertex with maximum modulus among all the vertices ofC−. Firstly, we delete an edge in each block such that no two deleted edges of any two blocks have a common vertex inC− and the deleted edge in a block which has v is not incident on this vertex. Thus, we obtain a subgraph ofC− such that its each block is an edge. Then by the same discussion as of equation (4.2) in Lemma 4.1, we obtain a cactus with at least one block as a cycle and at least one block as an edge which is infect a star K ,q with edges among the pendent vertices having di erent end points.
Since n ≡ (mod ) and n = p + q + , where n = |V+ ∪ V−| = |C+ ∪C−|, p + = |V+| = |C+| and q + = |V−| = |C−|. Therefore, either both p and q are even or odd.  .17), λ min (C (p, q) c ) ≤ λ min (C c ), where n = p + q + > and n ≡ (mod ). Similarly, it also can be prove for all other possibilities. Now nally we prove that there does not exist any vertex in V+ such that its value given by X is zero and E has exactly one edge. Firstly, among the vertices of C (p, q), we prove that v = u and v = u are unique ones in C+ and, v = v and v = v are unique ones in C− with maximum and minimum modulus, respectively. For this, we will show ≤ X < X < X < X and X < X < X < X < . By Lemma 4.2, we have X , X , X non negative and X , X , X , X , X negative values in the rst eigenvector X of C (p, q) c . By (2.5), λ (X − X ) = −(p − )X − X − X < , (λ + )(X − X ) = −(X − X ) < , and λ (X − X ) = X < ⇒ X − X > , X − X > and X − X > . Thus ≤ X < X < X < X . (4.18) Similarly, λ (X − X ) = X + X + (q − )X < , (λ + )(X − X ) = −X + X < and λ (X − X ) = −X < ⇒ X − X > , X − X > and X − X > . Thus If any one of the vertices v , v and v has value zero assigned by X, then by (4.18) X = = X . Moreover, by (2.5), we have X = = X , which is a contradiction to the construction of V− and C−. If the value of the vertex v labeled by X is zero, then by (2.5), λ (X − X ) = ⇒ X = X which is a contradiction to (4.19) (i.e. X is a unique one in C−). Consequently, X , X , X and X are non zero positive values of X. Thus, v ∉ V+ such that Xv = . By (4.4), (4.5), (4.8), (4.9) and the above discussion, we have C+ =C+ = B (p), C− =C− = B (q) and E has only one edge u v = v v in B (p, q). Similarly, we can prove for B (p, q) and B (p, q). This complete the proof.
Similarly, we can prove the following result: Lemma 4.4. Suppose that C is a cactus graph of order n = p + q + ≥ such that C c ∈ Ω c ,n . (a) If n ≡ (mod ), then either λ min (C (p, q) c ) < λ min (C c ) or λ min (C (p, q) c ) < λ min (C c ). (b) If n ≡ (mod ), then λ min (C (p, q) c ) < λ min (C c ).

Conclusions
Petrović et al. [23] explored a unique cactus as a minimizing graph from the class of cacti such that the order of each cactus is n. But, it is noted that the complement of the proposed minimizing graph is disconnected. In this paper, we characterize the minimizing graphs in a collection of connected graphs such that the complement of each graph of order n is a cactus with the condition that either its each block is only an edge or it has at least one block which is an edge and at least one block which is a cycle. However, the problem is still open to characterize the minimizing graphs in a collection of connected graphs whose complements are in the complete class of cacti (each block of a cactus is only an edge, at least one block is an edge and at least one block is a cycle, or each block is a cycle).