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Upper bounds for the global cyclicity index

  • José Luis Palacios EMAIL logo
Published/Copyright: July 16, 2024
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Open Mathematics
From the journal Open Mathematics Volume 22 Issue 1

Abstract

We find new upper bounds for the global cyclicity index, a variant of the Kirchhoff index, and discuss the wide family of graphs for which the bounds are attained.

Keywords: Kirchhoffian indices; reciprocal topological indices
MSC 2010: 05C92; 92E10

1 Introduction

Let G = ( V , E ) be a simple, connected, undirected graph, where V = { 1 , 2 , … , n } is the set of vertices and E the set of edges. We denote by R i j the effective resistance between vertices i and j when the graph is thought as an electric network, where every edge is a unit resistor, and a battery is placed between vertices i and j so that the voltage at one end is equal to 0 and at the other end is equal to 1. For more details on graph theory, we refer the reader to [1]; for the electric networks and their interaction with probability, a good reference is [2].

About three decades ago, Klein and Randić introduced in [3] the Kirchhoff index

K f ( G ) = ∑ i < j R i j

and opened the doors for a multitude of works related to this descriptor and others that combine the effective resistances with other parameters of the graph. Among these, we can mention the multiplicative degree-Kirchhoff index

K f * ( G ) = ∑ i < j d i d j R i j ,

introduced in [4], the additive degree-Kirchhoff index

K f + ( G ) = ∑ i < j ( d i + d j ) R i j ,

put forward in [5] (see [6] for some interplay between these Kirchhoffian indices), the mixed degree-Kirchhoff index

K ˆ f ( G ) = ∑ i < j d i d j + d j d i R i j ,

discussed in [7], and the global cyclicity index

C ( G ) = ∑ ( i , j ) ∈ E 1 R i j − ∣ E ∣ ,

introduced in [8].

In [9], using majorization, Yang found the bounds

(1) ∣ E ∣ ( ∣ E ∣ − n + 1 ) n − 1 ≤ C ( G ) ≤ n ( ∣ E ∣ − n + 1 ) 2 ,

where both equalities are attained by trees, for which C ( G ) = 0 , and by the complete graph K n . In [10], we extended these inequalities to general networks where the edges may not necessarily be unit resistors. In this article, using some general inequalities for topological descriptors, we prove again the lower bound in (1) and find new upper bounds for C ( G ) , in terms of the maximum and minimum effective resistances, where the equalities are attained by an extensive family of graphs that deserve some attention.

2 Results

For a general descriptor of the form

D p ( G ) = ∑ i = 1 N c i p ,

where c i s are the non-negative parameters of G , and for which m ≤ c i p ≤ M , we found in [11] the following bounds:

Lemma 1

For any descriptor D p ( G ) , we have

(2) N 2 ≤ D p ( G ) D − p ( G ) ≤ N 2 ( m + M ) 2 4 m M .

Both equalities are attained in case m = c i p = M for all 1 ≤ i ≤ N . Also, the right equality is attained if n is even and the first n 2 of the c i p s are equal to m and the other n 2 of the c i p s are equal to M .

Also, if n is odd, we have

(3) N 2 ≤ D p ( G ) D − p ( G ) ≤ N 2 ( M + m ) 2 − ( M − m ) 2 4 m M .

Both equalities are attained in case m = c i p = M for all 1 ≤ i ≤ N . Also, the right equality is attained if the first n − 1 2 of the c i p s are equal to m, the last n − 1 2 of the c i p s are equal to M, and the middle c i p is either m or M.

The left inequalities are a consequence of the arithmetic-harmonic-mean inequality. The right inequalities are shown using Schweitzer inequality, originally found in [12], and Lupaş inequality, a refinement of Schweitzer inequality proved in [13].

Given an edge in E , let us define its edge resistance as the effective resistance between the endpoints of that edge. Now, let us consider the auxiliary descriptor

C * ( G ) = ∑ ( i , j ) ∈ E 1 R i j .

It is well known (Foster’s first formula, [14]) that its reciprocal descriptor satisfies

C − 1 * ( G ) = ∑ ( i , j ) ∈ E R i j = n − 1 .

Let m = min ( i , j ) ∈ E R i j , M = max ( i , j ) ∈ E R i j . Then, applying the lemma, we have

(4) ∣ E ∣ 2 n − 1 ≤ C * ( G ) ≤ ∣ E ∣ 2 ( m + M ) 2 4 ( n − 1 ) m M ,

where both equalities are attained if all edge resistances are equal. Also, the right equality is attained if ∣ E ∣ is even and ∣ E ∣ 2 edge resistances are equal to m and the other ∣ E ∣ 2 are equal to M . Also, if ∣ E ∣ is odd, we have

(5) ∣ E ∣ 2 n − 1 ≤ C * ( G ) ≤ ∣ E ∣ 2 ( m + M ) 2 − ( M − m ) 2 4 ( n − 1 ) m M ,

Subtracting ∣ E ∣ to the left terms of Inequalities (4) and (5), we obtain

(6) ∣ E ∣ ( ∣ E ∣ − n + 1 ) n − 1 ≤ C ( G ) ,

which is the same left inequality in (1). Thus, we will focus only on the right inequalities: subtracting ∣ E ∣ to the right sides of (4) and (5), we have the following.

Theorem 1

For all G, we have

(7) C ( G ) ≤ ∣ E ∣ ∣ E ∣ ( m + M ) 2 4 ( n − 1 ) m M − 1 ,

where the equality is attained if all edge resistances are equal, or if  ∣ E ∣ is even, and ∣ E ∣ 2 of the edge resistances are equal to m and the other ∣ E ∣ 2 are equal to M.

Also, if ∣ E ∣ is odd, we have

(8) C ( G ) ≤ ∣ E ∣ 2 ( M + m ) 2 − ( M − m ) 2 4 ( n − 1 ) M m − ∣ E ∣ ,

where the equality is attained if all the edge resistances are equal, or if  ⌊ ∣ E ∣ 2 ⌋ of them are equal to m and the rest are equal to M, or if ⌈ ∣ E ∣ 2 ⌉ of them are equal to m and the rest are equal to M.

We want to give sufficient conditions for the cases where the Equalities in (7) and (8) are attained. As far back as 1961, Foster noted in [15] that “If the network is symmetrical with respect to its branches,” then all effective resistance of the edges are equal, and their value is

(9) R i j = n − 1 ∣ E ∣ , for all ( i , j ) ∈ E .

The sentence in the aforementioned quotes deserves some analysis. There are two possible edge symmetries in a graph: the stronger, perhaps the one espoused by Foster, is called arc transitivity, where for every two edges ( a , b ) and ( c , d ) , of the graph, there is an automorphism f such that a = f ( c ) and b = f ( d ) . There is a weaker form of edge symmetry, which is the edge transitivity, where for every pair of edges ( a , b ) and ( c , d ) , there is an automorphism f such that the sets { c , d } and { f ( a ) , f ( b ) } are the same. Obviously, arc transitivity implies edge transitivity, but the opposite is not true. For example, the star graph is edge-transitive but not arc-transitive. There are complex examples of edge-transitive and regular graphs, which are not arc-transitive [16], and even edge-transitive and vertex-transitive graphs, where this latter notion in italics means that every vertex can be mapped into any other by a suitable graph automorphism, which are not arc-transitive [17]. Now, even with the weaker form of symmetry, it is simple to see that the value of the effective resistance is constant over all edges. Indeed, since the automorphism preserves adjacencies, R a b = R f ( a ) f ( b ) , and R f ( a ) f ( b ) is either R c d or R d c , but these latter two values are the same, by the symmetry of the resistance distance, so R a b = R c d for any two edges ( a , b ) and ( c , d ) .

Edge transitivity is a sufficient but not necessary condition for (9) to hold: we could take several copies of the same edge-transitive graph and conjoin them so that any two of the copies have only cut-points as common vertices. For example, think of a 5-vertex graph made of two triangles where one vertex of one triangle is identified with one vertex of the other triangle; in this 5-vertex graph, the effective resistances of all edges are equal to 2 3 , but the graph is not edge-transitive. It would be interesting to find necessary and sufficient conditions for a graph to have a single value for all its edge resistances.

Now, let us look at a family of graphs that have exactly two different values for the effective resistances of its edges. If we consider a k -vertex linear graph attached to an r -cycle, we will obtain a graph where half the resistances of the edges are equal to m and the other half equal to M , as long as we take r = k . If we take r = k − 1 , we will obtain the case where ⌈ n 2 ⌉ of the edges have resistance m and the other ⌊ n 2 ⌋ edges have resistance M ; finally, if r = k + 1 , we obtain the case where ⌈ n 2 ⌉ of the edges have resistance M and the other ⌊ n 2 ⌋ edges have resistance m .

More involved examples can be found from conjoining, at a single point, edge-transitive graphs with distinct values of their edge resistances, for example, conjoining a complete graph K s with a cycle with s ( s − 1 ) 2 edges, etc.

Again, it would be interesting to know necessary and sufficient conditions for a graph to have exactly two different values of its edge resistances, in the same way that there are conditions under which a graph has only two non-zero Laplacian (or normalized Laplacian) eigenvalues [18,19].

  1. Funding information: The author thanks the Department of Electrical and Computer Engineering of the University of New Mexico for the support in covering the publishing costs of this article.

  2. Author contributions: The author confirms the sole responsibility for the conception of the study, presented results, and preparation of the manuscript.

  3. Conflict of interest: The author declares that there is no conflict of interest in regard to this article.

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Received: 2023-12-14
Revised: 2024-04-16
Accepted: 2024-04-23
Published Online: 2024-07-16

© 2024 the author(s), published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

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https://doi.org/10.1515/math-2024-0016
Keywords for this article
Kirchhoffian indices; reciprocal topological indices
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  74. Construction of a class of half-discrete Hilbert-type inequalities in the whole plane
  75. Analysis of two-grid method for second-order hyperbolic equation by expanded mixed finite element methods
  76. Note on stability estimation of stochastic difference equations
  77. Trigonometric integrals evaluated in terms of Riemann zeta and Dirichlet beta functions
  78. Purity and hybridness of two tensors on a real hypersurface in complex projective space
  79. Classification of positive solutions for a weighted integral system on the half-space
  80. A quasi-reversibility method for solving nonhomogeneous sideways heat equation
  81. Higher-order nonlocal multipoint q-integral boundary value problems for fractional q-difference equations with dual hybrid terms
  82. Noetherian rings of composite generalized power series
  83. On generalized shifts of the Mellin transform of the Riemann zeta-function
  84. Further results on enumeration of perfect matchings of Cartesian product graphs
  85. A new extended Mulholland's inequality involving one partial sum
  86. Power vector inequalities for operator pairs in Hilbert spaces and their applications
  87. On the common zeros of quasi-modular forms for Γ+0(N) of level N = 1, 2, 3
  88. One special kind of Kloosterman sum and its fourth-power mean
  89. The stability of high ring homomorphisms and derivations on fuzzy Banach algebras
  90. Integral mean estimates of Turán-type inequalities for the polar derivative of a polynomial with restricted zeros
  91. Commutators of multilinear fractional maximal operators with Lipschitz functions on Morrey spaces
  92. Vector optimization problems with weakened convex and weakened affine constraints in linear topological spaces
  93. The curvature entropy inequalities of convex bodies
  94. Brouwer's conjecture for the sum of the k largest Laplacian eigenvalues of some graphs
  95. High-order finite-difference ghost-point methods for elliptic problems in domains with curved boundaries
  96. Riemannian invariants for warped product submanifolds in Q ε m × R and their applications
  97. Generalized quadratic Gauss sums and their 2mth power mean
  98. Euler-α equations in a three-dimensional bounded domain with Dirichlet boundary conditions
  99. Enochs conjecture for cotorsion pairs over recollements of exact categories
  100. Zeros distribution and interlacing property for certain polynomial sequences
  101. Random attractors of Kirchhoff-type reaction–diffusion equations without uniqueness driven by nonlinear colored noise
  102. Study on solutions of the systems of complex product-type PDEs with more general forms in ℂ2
  103. Dynamics in a predator-prey model with predation-driven Allee effect and memory effect
  104. A note on orthogonal decomposition of 𝔰𝔩n over commutative rings
  105. On the δ-chromatic numbers of the Cartesian products of graphs
  106. Binomial convolution sum of divisor functions associated with Dirichlet character modulo 8
  107. Commutator of fractional integral with Lipschitz functions related to Schrödinger operator on local generalized mixed Morrey spaces
  108. System of degenerate parabolic p-Laplacian
  109. Stochastic stability and instability of rumor model
  110. Certain properties and characterizations of a novel family of bivariate 2D-q Hermite polynomials
  111. Stability of an additive-quadratic functional equation in modular spaces
  112. Monotonicity, convexity, and Maclaurin series expansion of Qi's normalized remainder of Maclaurin series expansion with relation to cosine
  113. On k-prime graphs
  114. On the existence of tripartite graphs and n-partite graphs
  115. Classifying pentavalent symmetric graphs of order 12pq
  116. Almost periodic functions on time scales and their properties
  117. Some results on uniqueness and higher order difference equations
  118. Coding of hypersurfaces in Euclidean spaces by a constant vector
  119. Cycle integrals and rational period functions for Γ0+(2) and Γ0+(3)
  120. Degrees of (L, M)-fuzzy bornologies
  121. A matrix approach to determine optimal predictors in a constrained linear mixed model
  122. On ideals of affine semigroups and affine semigroups with maximal embedding dimension
  123. Solutions of linear control systems on Lie groups
  124. A uniqueness result for the fractional Schrödinger-Poisson system with strong singularity
  125. On prime spaces of neutrosophic extended triplet groups
  126. On a generalized Krasnoselskii fixed point theorem
  127. On the relation between one-sided duoness and commutators
  128. Non-homogeneous BVPs for second-order symmetric Hamiltonian systems
  129. Erratum
  130. Erratum to “Infinitesimals via Cauchy sequences: Refining the classical equivalence”
  131. Corrigendum
  132. Corrigendum to “Matrix stretching”
  133. Corrigendum to “A comprehensive review of the recent numerical methods for solving FPDEs”
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