Upper bounds for the global cyclicity index

: We ﬁ nd new upper bounds for the global cyclicity index, a variant of the Kirchho ﬀ index, and discuss the wide family of graphs for which the bounds are attained.

V E , be a simple, connected, undirected graph, where { } = V n 1, 2, …, is the set of vertices and E the set of edges.We denote by R ij 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 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 introduced in [4], the additive degree-Kirchhoff index put forward in [5] (see [6] for some interplay between these Kirchhoffian indices), the mixed degree-Kirchhoff index discussed in [7], and the global cyclicity index In [9], using majorization, Yang found the bounds 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.

Results
For a general descriptor of the form where c i s are the non-negative parameters of G, and for which ≤ ≤ m c M i p , we found in [11] the following bounds: Both equalities are attained in case . 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) . 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 It is well known (Foster's first formula, [14]) that its reciprocal descriptor satisfies . Then, applying the lemma, we have 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 (5) Subtracting | | E to the left terms of Inequalities ( 4) and ( 5), we obtain 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 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 where the equality is attained if all the edge resistances are equal, or if of them are equal to m and the rest are equal to M, or if 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 , for all , .
ij (9) 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, , 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.
Upper bounds for the global cyclicity index  3 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 ⌈ ⌉ 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].
b , and ( ) ( ) R f a f b is either R cd or R dc , but these latter two values are the same, by the symmetry of the resistance distance, so =

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.