For a reaction network N with species set S , a log-parametrized (LP) set is a non-empty set of the form E(P, x * ) = {x ∈ R S > | log x − log x * ∈ P ⊥ } where P (called the LP set's flux subspace) is a subspace of R S , x * (called the LP set's reference point) is a given element of R S > , and P ⊥ (called the LP set's parameter subspace) is the orthogonal complement of P . A network N with kinetics K is a positive equilibria LP (PLP) system if its set of positive equilibria is an LP set, i.e., E+(N , K) = E(PE, x * ) where PE is the flux

For a reaction network N with species set S , a log-parametrized (LP) set is a non-empty set of the form E(P, x * ) = {x ∈ R S > | log x − log x * ∈ P ⊥ } where P (called the LP set's flux subspace) is a subspace of R S , x * (called the LP set's reference point) is a given element of R S > , and P ⊥ (called the LP set's parameter subspace) is the orthogonal complement of P . A network N with kinetics K is a positive equilibria LP (PLP) system if its set of positive equilibria is an LP set, i.e., E + (N , K) = E(P E , x * ) where P E is the flux subspace and x * is a given positive equilibrium. Analogously, it is a complex balanced equilibria LP (CLP) system if its set of complex balanced equilibria is an LP set, i.e., Z + (N , K) = E(P Z , x * ) where P Z is the flux subspace and x * is a given complex balanced equilibrium. An LP kinetic system is a PLP or CLP system. This paper studies concentration robustness of a species on subsets of equilibria, i.e., the invariance of the species concentration at all equilibria in the subset. We present the "species hyperplane criterion", a necessary and sufficient condition for absolute concentration robustness (ACR), i.e., invariance at all positive equilibria, for a species of a PLP system. An analogous criterion holds for balanced concentration robustness (BCR), i.e., invariance at all complex balanced equilibria, for species of a CLP system. These criteria also lead to interesting necessary properties of LP systems with concentration robustness. Furthermore, we show that PLP and CLP power law systems with Shinar-Feinberg reaction pairs in species X, i.e., their rows in the kinetic order matrix differ only in X, in a linkage class have ACR and BCR in X, respectively. This leads to a broadening of the "low deficiency building blocks" framework introduced by Fortun and Mendoza (2020) to include LP systems of Shinar-Feinberg type with arbitrary deficiency. Finally, we apply our results to species concentration robustness in LP systems with poly-PL kinetics, i.e., sums of power law kinetics, including a refinement of a result on evolutionary games with poly-PL payoff functions and replicator dynamics by Talabis et al (2020).

The interplay of groups and graphs has been a subject of interest by mathematics researchers nowadays. One particular instance is the identity graph of a group introduced by Kandasamy [5]. Moreover, the concept of a central graph of any graph is widely used by many graph theorists. The central graph of a graph G denoted by C(G) can be obtained by subdividing the edge of G exactly once and joining all the nonadjacent vertices of G in C(G). In this paper, we construct the central graph of the identity graph of finite cyclic group and investigate some of its graph properties.