Some improved bounds on two energy-like invariants of some derived graphs

Lemma 2.1

[27, 29] If G is an r-regular graph of order n with m edges, then

1. If the L-spectrum of G is Sp(L(G)) = {μ₁ …, μ_n}, then the L-spectrum of 𝓡(G)

   Sp(L(R(G)))=2(m−n),(r+2+μi)±(r+2+μi)2−12μi2(i=1,2,…,n).

2. If the Q-spectrum of G is Sp(Q(G)) = {q₁, …, q_n}, then the Q-spectrum of 𝓡(G)

   Sp(Q(R(G)))=2(m−n),(r+2+qi)±(r+2+qi)2−4(2r+qi)2(i=1,2,…,n).

   The 𝓠-graph [2] of G, denoted by 𝓠(G), is the graph derived from G by plugging a vertex w_i to every edge e_i = uv of G and by adding a new edge between two new vertices whenever these new vertices lie on adjacent edges of G. If G is an r-regular graph of order n with m edges, then L-polynomial [27, 28] and Q-polynomial [29] of 𝓠(G) are, respectively,

   ψ(L(Q(G)),x)=x(x−2r−2)m−n(x−r−2)∏i=1n−1[(x−r)(x−2−μi)−2r+μi] (7)

   and

   ψ(Q(Q(G)),x)=(x−2r+2)m−n[(x−r)(x−4r+2)−2r]∏i=2n[(x−r)(x−2r+2−qi)−qi]. (8)

   Similarly, from (7) and (8), one obtains the following lemma easily.

Lemma 2.2

[27, 29] If G is an r-regular graph of order n with m edges, then

1. If the L-spectrum of G is Sp(L(G)) = {μ₁, …, μ_n}, then the L-spectrum of 𝓠(G)

   Sp(L(Q(G)))=(2r+2)(m−n),(r+2+μi)±(r+2+μi)2−4μi(r+1)2(i=1,2,…,n).

2. If the Q-spectrum of G is Sp(Q(G)) = {q₁, …, q_n}, then the Q-spectrum of 𝓠(G)

   Sp(Q(Q(G)))=(2r−2)(m−n),(3r−2+qi)±(3r−2+qi)2−4r(2r−2+qi)+4qi2,

   where i = 1, …, n.

Lemma 2.3

[26] If G is an r-regular graph of order n, then

where both equalities hold if and only if G is the complete graph K_n.

Lemma 2.4

[30] If G is any graph of order n, with at least one edge, then μ₁ = μ₂ = ⋯ = μ_n−1 if and only if G is the complete graph K_n.

The following lemma for Q-spectrum is analogous to above Lemma 2.4 for L-spectrum. By Theorem 3.6 in [1], one obtains the following lemma easily.

Lemma 2.5

If G is a graph of order n, with at least one edge, then q₂ = q₃ = ⋯ = q_n if and only if G is the complete graph K_n.

The following lemma comes from [31], which is called the Ozeki’s inequality.

Lemma 2.6

[31] Let ξ = (a₁, …, a_n) and η = (b₁, …, b_n) be two positive n-tuples with 0 < p ≤ a_i ≤ P and 0 < q ≤ b_i ≤ Q, where i = 1, …, n. Then

It is a remarkable fact that a refinement of Ozeki’s inequality was obtained by Izumino et al.[32] below.

Lemma 2.7

[32] Let ξ = (a₁, …, a_n) and η = (b₁, …, b_n) be two n-tuples with 0 ≤ p ≤ a_i ≤ P, 0 ≤ q ≤ b_i ≤ Q and PQ ≠ 0, where i = 1, …, n. Take α = p/P and β = q/Q. If (1 + α)(1 + β) ≥ 2, then (9) still holds.

Remark that if G is 1-regular, then G is isomorphic to n2 K₂. For avoiding the triviality, we always suppose that r ≥ 2 for an r-regular graph. For an (r₁, r₂)-semiregular graph G, G is isomorphic to n3 P₃ whenever r₁+r₂ = 3. Next we also suppose that r₁+r₂ ≥ 4 for an (r₁, r₂)-semiregular graph throughout this paper. In addition, it is well known [2, 3] that the largest Laplacian eigenvalue μ₁ ≤ 2r and largest signless Laplacian eigenvalue q₁ = 2r for an r-regular graph. From Lemma 3.3 in [23], we also see that μ₁ = r₁+r₂ for an (r₁, r₂)-semiregular graph.

Theorem 3.1

If G is an r-regular graph of order n with m edges, then

1. LEL(R(G))≤n(r−2)22+r+2+(n−1)r+2+nrn−1+23n−1LEL(G), (10)

   where the equality holds in (10) if and only if G is the complete graph K_n.

2. LEL(R(G))≥n(r−2)22+r+2+(n−1)34(r+2)+nrn−1+23n−1LEL(G). (11)

Proof

Suppose that Sp(L(G)) = {μ₁, μ₂, …, μ_n} is the L-spectrum of G. Then from (1) and the (i) in Lemma 2.1, one gets

Notice that ∑i=1n−1 μ_i = 2m = nr. Applying the Cauchy-Schwarz inequality, one obtains

where above equality holds if and only if μ₁ = μ₂ = ⋯ = μ_n−1. It follows from Lemma 2.4 that G is the complete graph K_n. Hence the proof of the (i) is completed.

Now we prove the (ii). Assume that ai=r+2+μi+23μi and b_i = 1, i = 1, …, n − 1. Take P = 3r+2+26r,p=r+2 and Q = q = 1. Since 0 ≤ μ_i ≤ 2r, then 0 < p ≤ a_i ≤ P, 0 < q ≤ b_i ≤ Q and

By Lemma 2.6, we obtain

From (12), one obtains the required result (ii).□

Corollary 3.2

If G is an r-regular graph of order n with m edges, then

1. LEL(R(G))≤n(r−2)22+r+2+(n−1)r+2+nrn−1+23(r+1+(n−2)(nr−r−1))n−1,

   where above equality holds if and only if G is the complete graph K_n.

2. LEL(R(G))>n(r−2)22+r+2+(n−1)34(r+2)+nrn−1+23n−1nrr+1.

Proof

Theorem 3.1 (i) and Lemma 2.3 together imply (i) in the corollary. Again, from Lemma 2.3 and (11), one gets

Suppose that the equality in (13) holds. From Lemma 2.3, we have G is the complete graph K_n. But for the complete graph K_n, the inequality (10) implies that the equality is false in (11). This completes the proof.□

Remark 1

Given an r-regular graph G of order n, Pirzada et al. [23] proved that

where the equality on the right of (14) holds if and only if G is the complete graph K₂. Notice that these bounds in Corollary 3.2 improve those in (14). In fact, by direct computation, we have

which implies that the upper bound in Corollary 3.2 is an improvement on that in (14). For the lower bound, it is easy to see that

Hence the lower bound in Corollary 3.2 is also an improvement on that in (14).

Next we consider the Laplacian-energy-like invariant of 𝓠-graph of a regular graph.

Theorem 3.3

If G is an r-regular graph of order n with m edges, then

1. LEL(Q(G))≤n(r−2)22r+2+r+2+(n−1)r+2+nrn−1+2r+1n−1LEL(G), (15)

   where the equality holds in (15) if and only if G is the complete graph K_n.

2. LEL(Q(G))>n(r−2)22r+2+r+2+(n−1)r+2+nrn−1+2r+1n−1LEL(G)−34r. (16)

Proof

Assume that Sp(L(G)) = {μ₁, …, μ_n} is L-spectrum of G. By the (i) in Lemma 2.2 and (1), one has

Notice that ∑i=1n−1 μ_i = 2m = nr. Applying the Cauchy-Schwarz inequality, one obtains

with the equality holding if and only if μ₁ = μ₂ = ⋯ = μ_n−1. By Lemma 2.4, G is the complete graph K_n. The proof of the (i) is completed.

Next we prove the (ii). Assume that ai=r+2+μi+2(r+1)μi and b_i = 1, where i = 1, …, n − 1. Take P=3r+2+22r(r+1),p=r+2 and Q = q = 1. Since 0 ≤ μ_i ≤ 2r and P=3r+2+22r(r+1)≤7r+2. Then 0 < p ≤ a_i ≤ P, 0 < q ≤ b_i ≤ Q and

From (17) and Lemma 2.6, one has

From (17), one obtains the required result (ii).□

By Theorem 3.3, we obtain Corollary 3.4 immediately.

Corollary 3.4

If G is an r-regular graph of order n with m edges, then

1. LEL(Q(G))≤(n−1)r+2+nrn−1+2r+1(r+1+(n−2)(nr−r−1))n−1+n(r−2)22r+2+r+2,

   where the equality holds if and only if G is the complete graph K_n.

2. LEL(Q(G))>n(r−2)22r+2+r+2+(n−1)(3nn−1+14)r+2.

Remark 2

Given an r-regular graph G, Pirzada et al. [23] proved that

where the equality on the right of (18) holds if and only if G is the complete graph K₂. Since (3nn−1+14)r>r, then the lower bound in Corollary 3.4 is an improvement on that in (18). For the upper bound, one has

Hence, the upper bound in Corollary 3.4 is also an improvement on that in (18).

We finally consider the LEL of line graph of an (r₁, r₂)-semiregular graph. Pirzada et al. [23] presented the following an upper bound on LEL of line graph 𝓛(G) for an (r₁, r₂)-semiregular graph G, that is,

Next we shall give a lower bound on LEL of its line graph 𝓛(G).

Theorem 3.5

If G is an (r₁, r₂)-semiregular graph of order n with m edges, then

Proof

Suppose that Sp(L(G)) = {μ₁, …, μ_n} is the L-spectrum of G. Since μ₁ = r₁ + r₂ and μ_n = 0, then from (1) and (3), one gets

Now, assume that ai=r1+r2−μi and b_i = 1, i = 2, …, n − 1. Take P=r1+r2, p = 0 and Q = q = 1. Obviously, 0 ≤ p ≤ a_i ≤ P, 0 ≤ q ≤ b_i ≤ Q, PQ ≠ 0 and (1 + p/P)(1 + q/Q) ≥ 2. From Lemma 2.7, we have

which yields the required result as m = nr₁r₂/(r₁+r₂).□

Theorem 4.1

If G is an r-regular graph of order n with m edges, then

1. IE(R(G))≤n(r−2)22+3r+2+4r+(n−1)2n−3n−1r+23n−4n−1r+2, (19)

   where the equality holds if and only if G is the complete graph K_n.

2. IE(R(G))>n(r−2)22+3r+2+4r+(n−1)(2n−3n−1−2−32)r+2(3n−4n−1−3−222)r+2. (20)

Proof

Assume that Sp(Q(G)) = {q₁, …, q_n} is the Q-spectrum of G. Notice that q₁ = 2r as G is r-regular. Then from (2) and the (ii) in Lemma 2.1, we obtain, by a simple calculation,

Clearly, ∑i=2n q_i = 2m − 2r = (n − 2)r. Applying the Cauchy-Schwarz inequality, one obtains

From (21), we obtain the desired upper bound (19). Moreover, above equality occurs if and only if q₁ = 2r and q₂ = q₃ = ⋯ = q_n. Thus by Lemma 2.5, G is the complete graph K_n. The proof of the (i) is completed.

Next we prove the (ii). Assume that ai=r+2+qi+22r+qi and b_i = 1, i = 2, …, n. Take P = 3r+2+4r,p=r+2+22r and q = Q = 1. Since 0 ≤ q_i ≤ 2r, then 0 < p ≤ a_i ≤ P, 0 < q ≤ b_i ≤ Q. By a simple computation, one has

Then by Lemma 2.6, one has

Similarly, assume that ai=2r+qi and b_i = 1, i = 2, …, n. Take P=2r,p=2r and Q = q = 1. Since 0 ≤ q_i ≤ 2r, then 0 < p ≤ a_i ≤ P, 0 < q ≤ b_i ≤ Q. Again by Lemma 2.6, one has

Hence,

which, along with (21), implies the required result (ii).□

Next we shall consider IE of 𝓠-graph for regular graphs.

Theorem 4.2

If G is an r-regular graph of order n with m edges, then

1. IE(Q(G))≤n(r−2)22r+2+5r−2+4r(r−1)+(n−1)4n−5n−1r+23n−4n−1r(r−1)−2, (22)

   where the equality holds in (22) if and only if G is the complete graph K_n.

2. IE(Q(G))>n(r−2)22r+2+5r−2+4r(r−1)+(n−1)(4n−5n−1−14)r+2(3n−4n−1−3−222)r(r−1)−2. (23)

Proof

Assume that Sp(Q(G)) = {q₁, …, q_n} is the Q-spectrum of G. Notice that q₁ = 2r as G is r-regular. Then from (2) and the (ii) in Lemma 2.2, it is easy to see that, by a simple calculation,

Clearly, ∑i=2n q_i = 2m − 2r = (n − 2)r. Applying the Cauchy-Schwarz inequality, one obtains

which, along with (24), implies the desired upper bound. Moreover, above equality occurs if and only if q₁ = 2r and q₂ = q₃ = ⋯ = q_n. Thus from Lemma 2.5, G is the complete graph K_n. The proof of the (i) is completed.

Now we prove the (ii). Assume that ai=3r−2+qi+2r(2r−2+qi)−qi and b_i = 1, where i = 2, …, n. Take P=5r−2+4r(r−1),p=3r−2+22r(r−1) and Q = q = 1. Since 0 ≤ q_i ≤ 2r, then 0 < p ≤ a_i ≤ P, 0 < q ≤ b_i ≤ Q. By a simple computation, one has

Then by Lemma 2.6, one has

Similarly, suppose that ai=r(2r−2+qi)−qi and b_i = 1, i = 2, 3, …, n. Take P=2r(r−1),p=2r(r−1) and Q = q = 1. Since 0 ≤ q_i ≤ 2r, then 0 < p ≤ a_i ≤ P, 0 < q ≤ b_i ≤ Q. Again, from Lemma 2.6, one obtains

Hence,

It follows from (24) that the (ii) holds.□

We finally consider the incidence energy of line graph for an (r₁, r₂)-semiregular graph. In [25], an upper bound on IE of line graph 𝓛(G) for an (r₁, r₂)-semiregular graph G was obtained as follows:

Below one gives a lower bound for IE of its line graph 𝓛(G).

Theorem 4.3

If G is an (r₁, r₂)-semiregular graph with n vertices and m edges, then

Proof

Suppose that Sp(Q(G)) = {q₁, …, q_n} is the Q-spectrum of G. Notice that q₁ = r₁ + r₂ and q_n = 0 as G is bipartite. Then from (2) and (4)), one gets