Introduction

Let G be a simple graph with the vertex set V(G) and the edge set E(G). If the vertices u, v ∈ V(G) are adjacent, then the edge connecting them is denoted by uv. The number of edges incident to a vertex v is the degree of that vertex, and is denoted by d(v). The minimum and maximum vertex degrees are denoted by $\mathrm{\delta }$ and ∆, respectively.

Let V(G) = {v₁., v₂, . . . , v._n}. Then the adjacency matrix A(G) = [a_ij] of the graph G is the symmetric matrix of order n, whose elements are (Cvetković et al, 2010):

If the eigenvalues of A(G) are λ₁, . . . , λ_n, then the (ordinary) energy of the graph G is defined as

The theory of graph energy is nowadays elaborated in due detail (Li et al, 2012; Ramane, 2020).

In the chemical and mathematical literature, a variety of vertex-degree-based (VDB) graph invariants of the form

has been considered, where f is a suitably chosen function, with a property f(x, y) = f(y, x) (Kulli, 2020; Todeschini & Consonni, 2009). These are usually referred to as topological indices. Of these, we list here a few most popular and best studied ones (Table 1):

The parameters x and y (being vertex degrees) always satisfy the condition x ≥ 1 , y ≥ 1. Bearing this in mind, we immediately recognize that most VDB indices are either monotonically increasing (↑) or monotonically decreasing functions (↓) of the vertex degrees. Only a few such indices do not possess such a monotonicity property (∼).

It should be noted that for practically all VDB indices of type ↑ that exist in the literature, the condition f (x, y) ≥ 1 is satisfied for all values of x and y that occur for the edges of graphs.

Analogously, for practically all VDB indices of type ↓, 0 < f (x, y) ≤ 1 holds for all values of x and y.

Taking into account Eq (1) and (3) we introduce the VDB matrix

via

If its eigenvalues are µ₁, . . . , µ_n, then the energy pertaining to the VDB invariant, Eq. (3), is

For recent works on the investigation of this class of graph-spectral invariants see (Das et al, 2018; Gutman, 2020; Gutman, 2021; Gutman et al, 2022; Li & Wang, 2021; Shao et al, 2021).

Main results

A cycle of length p is a cycle consisting of (exactly) p vertices v₁, v₂, . . . , v_p, so that v_i and v_i+1 are adjacent for i = 1, 2, . . . , p − 1, and also v₁ and v_p are adjacent. As it is well known, a graph G is bipartite if and only if all its cycles (if any) are of even length. In this paper, we prove the results valid for bipartite graphs which do not possess cycles of a length divisible by 4. Let G be such a graph. Without loss of generality, we assume that G is connected.

Let the graph energy $\text{Ɛ}$ and the VDB energy be the quantities defined via Eqs. (2) and (5), and let be the function specified in Eq. (3). Let δ ad ∆ be the smallest and largest vertex degrees of G.

Theorem 1. Let G be a bipartite graph with no cycle of size divisible by 4. Then

holds for all VDB invariants in which the function f is monotonically increasing and f (x, y) ≥ 1 for all vertex degrees x and y. Equality on both sides holds if and only if G is a regular graph, in which case δ = ∆.

The examples of the VDB invariants for Theorem 1 are the above listed first and second Zagreb, forgotten, Sombor, and nirmala indices.

Theorem 2. Let G be a bipartite graph with no cycle of size divisible by 4. Then

holds for all VDB invariants in which the function f is monotonically decreasing and 0 < f (x, y) ≤ 1 for all vertex degrees x and y. Equality on both sides holds if and only if G is a regular graph.

The examples of the VDB invariants for Theorem 2 are the above listed Randi´c, sum-connectivity, harmonic, and modified Sombor indices, as well as the inverse degree.

A tree is a connected graph with no cycles. Therefore, Theorems 1 and 2 apply to trees. For any tree δ = 1, but Theorems 1 and 2 can be slightly strengthened.

Theorem 3. Let T be a tree with n ≥ 3 vertices. Then

holds for all V DB invariants in which the function f is monotonically increasing and

f (x, y) ≥ 1 for all x, y. Equality on the left-hand side holds if and only if n = 3.

Theorem 4. Let T be a tree with n ≥ 3 vertices. Then

holds for all VDB invariants in which the function f is monotonically decreasing and

0 < f (x, y) ≤ 1 for all x, y. Equality on the right-hand side holds if and only if n = 3.

In addition to trees, Theorems 1 and 2 are applicable to various classes of cycle- containing graphs. Of these, of particular interest may be the hexagonal systems (molecular graphs of benzenoid hydrocarbons) (Gutman & Cyvin, 1989). All their vertices are of degrees 2 and 3. The so-called catacondesned hexagonal systems (= hexagonal systems having no internal vertices) are known to possess only cycles of size 4p + 2. For these molecular graphs

or

depending on whether f (x, y) monotonically increases or decreases.

Hexagonal systems possessing internal vertices have cycles of size 4p, p = 3, 4, etc., and thus Theorems 1 and 2 are not applicable. We nevertheless conjecture that estimates (6) and (7) are valid for all hexagonal systems.

In order to prove the above theorems, we need an auxiliary result, stated below as Lemma 3.

Energy of a weighted bipartite graph

The main part of the results outlined in this section was reported in (Gutman et al, 2021). These are repeated here (in an abbreviated form) in order to maintain completeness. Also, a few errors committed in (Gutman et al, 2021) are corrected.

Let G be a bipartite graph with n vertices. Let G_w be obtained from G by associating weighs to its edges, so that wij is the weight of the edge ij. Then the characteristic polynomial of G_w is of the form (Cvetković et al, 2010)

whereas the energy of G_w satisfies the equality (Gutman, 1977), (Gutman, 2020), (Li et al, 2012)

Note that $\text{Ɛ}$(G_w) is a monotonically increasing function of any of the coefficients c(G_w,k).

According to the Sachs theorem (Cvetković et al, 2010).

where S_k(G_w) is the set of all Sachs graphs of G_w possessing exactly 2k vertices, and where σ is an element of S_2k(G_w), containing p(σ) components, of which c(σ) are cycles. The weight of the Sachs graph σ is equal to the product of the weights of its components.

If the isolated edge ij is a component of σ, then its weight is ${\text{w}}_{\text{ij}}^{\text{2}}$ . If a cycle Z is a component of σ, then its weight is the product of weights of the edges contained in Z.

Lemma 1. (Gutman et al, 2021) If the Sachs graph σ ∈ S_2k(G_w) $\ne$ ∅ does not contain cycles whose size is divisible by 4, then

(−1)^k (−1)^p(σ) 2^c(σ) > 0 .

Proof. The Sachs graph σ has p(σ) components. Let among them be r₀ ≥ 0 isolated edges, whose total number of vertices is 2r₀. Let σ contain r₁ ≥ 0 cycles, whose total number of vertices is 4x + 2 r₁ for some integer x. Thus, 2k = 2r₀ + 4x + 2 r₁.

Case 1: 2k is not divisible by 4. Then (−1)^k = −1 whereas r₀ + r₁ = p(σ) is odd.

Therefore, (−1)^k (−1)^p(σ) > 0 and the claim of Lemma 1 holds.

Case 2: 2k is divisible by 4. Then (−1)^k = +1 whereas r₀ + r₁ = p(σ) is even, implying, again, (−1)^k (−1)^p(σ) > 0.

Lemma 1 has the following noteworthy consequences:

Lemma 2.

(a) Let G_w be an edge-weighted bipartite graph whose all cycles (if any) have size not divisible by 4, and let the weights of all its edges be positive-valued. Then for any Sachs graph σ ∈ S_2k(G_w) $\ne$ ∅,

(−1)^k (−1)^p(σ) 2^c(σ) w(σ) > 0 .

(b) Therefore, because of Eq. (10), the coefficients c(G_w, k) in Eq. (8) are non-negative and are the monotonically increasing functions of the edge-weights.

(c) Therefore, because of Eq. (9), the energy of the graphs G_w is a monotonically increas- ing function of the edge-weights.

From Lemma 2(c), we obtain the result needed for our proofs:

Lemma 3. Let G_w be an edge-weighted bipartite graph whose all cycles (if any) have size not divisible by 4.

(a) If for all edges ij ∈ E(G_w), the condition w_ij ≥ 1 holds, then $\text{Ɛ}$ (G_w) ≥ $\text{Ɛ}$ (G).

If w_ij > 1 for at least one edge ij, then $\text{Ɛ}$ (Gw) > $\text{Ɛ}$ (G).

(b) If for all edges ij ∈ E(G_w), the condition w_ij ≤ 1 holds, then $\text{Ɛ}$ (G_w) ≤ $\text{Ɛ}$ (G).

If w_ij < 1 for at least one edge ij, then $\text{Ɛ}$ (Gw) < $\text{Ɛ}$ (G).

(c) If in both cases (a) and (b), w_ij = w holds for all edges ij ∈ E(G_w), then $\text{Ɛ}$ (G_w) = w $\text{Ɛ}$ (G).

Proof of Theorems 1-4

The adjacency matrix , Eq. (4), could be viewed as the ordinary adjacency matrix of an edge-weighted modification of the graph G. Therefore, if the condition f (dvi, dvj ) >

Therefore, if the condition f(dv_i,dv_j > 1) holds, and if f (x, y) is an increasing function for x ≥ 1 and y ≥ 1, then the lower bound of Theorem 1 follows by Lemma 3 if all f (x, y) are replaced by f (δ, δ).

The upper bound is obtained if all f (x, y) are replaced by f (∆, ∆).

The proof of Theorem 2 is analogous.

Theorems 3 and 4 are based on the fact that no tree with n ≥ 3 vertices is a regular graph. The only tree having two adjacent degree-one vertices is the two-vertex tree. Therefore, for trees with 3 or more vertices, the minimal (resp. maximal) value of f (x, y) is f (1, 2).

References

Cvetković, D., Rowlinson, P. & Simić, S. K. 2010. An Introduction to the Theory of Graph Spectra. Cambridge: Cambridge University Press. ISBN: 9780521134088.

Das, K.C., Gutman, I., Milovanović, I., Milovanović, E. & Furtula, B. 2018. Degree- based energies of graphs. Linear Algebra and its Applications, 554, pp.185-204. Available at: https://doi.org/10.1016/j.laa.2018.05.027.

Gutman, I. 1977. Acyclic systems with extremal Huckel π-electron energy. Theoretica Chimica Acta, 45, pp.79-87. Available at: https://doi.org/10.1007/BF00552542.

Gutman, I. 2020. Relating graph energy with vertex-degree-based energies. Vojnotehnički glasnik/Military Technical Courier, 68(4), pp.715-725. Available at: https://doi.org/10.5937/vojtehg68-28083.

Gutman, I. 2021. Comparing degree-based energies of trees. Contributions to Mathematics, 4, pp.1-5. Available at: https://doi.org/10.47443/cm.2021.0030.

Gutman, I. & Cyvin, S.J. 1989. Introduction to the theory of benzenoid hydrocarbons. Berlin: Springer. Available at: https://doi.org/10.5860/choice.27-4521.

Gutman, I., Monsalve, J. & Rada, J. 2022. A relation between a vertex-degree-based topological index and its energy. Linear Algebra and Its Applications, 636(March), pp.134-142. Available at: https://doi.org/10.1016/j.laa.2021.11.021.

Gutman, I., Redžepović, I. & Rada, J. 2021. Relating energy and Sombor energy. Contributions to Mathematics, 4, pp.41-44. Available at: https://doi.org/10.47443/cm.2021.0054.

Kulli, V. R. 2020. Graph indices. In: Pal, M., Samanta, S. & Pal, A. (Eds.), Handbook of Research on Advanced Applications of Graph Theory in Modern Society. Hershey (USA): IGI Global, pp.66–91. Available at: https://doi.org/10.4018/978-1-5225-9380-5.ch003.

Li, X., Shi, Y. & Gutman, I. 2012. Graph Energy. New York: Springer. Available at: https://doi.org/10.1007/978-1-4614-4220-2_1.

Li, X. & Wang, Z. 2021. Trees with extremal spectral radius of weighted adjacency matrices among trees weighted by degree-based indices. Linear Algebra and Its Applications, 620, pp.61-75. Available at: https://doi.org/10.1016/j.laa.2021.02.023.

Ramane, H.S. 2020. Energy of graphs. In: Pal, M., Samanta, S. & Pal, A. (Eds.), Handbook of Research on Advanced Applications of Graph Theory in Modern Society. Hershey (USA): IGI Global, pp.267-296. Available at: https://doi.org/10.4018/978-1-5225-9380-5.ch011.

Shao, Y., Gao, Y., Gao, W. & Zhao, X. 2021. Degree-based energies of trees. Linear Algebra and Its Applications, 621, pp.18-28. Available at: https://doi.org/10.1016/j.laa.2021.03.009.

Todeschini, R. & Consonni, V. 2009. Molecular Descriptors for Chemoinformatics. Weinheim: Wiley-VCH. ISBN: 978-3-527-31852-0.

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