# Vertex and Mixed $k$-Diameter Component Connectivity

Adam Buzzard  Moravian College (Undergraduate) 1200 Main Street Bethlehem, PA
18018<EMAIL_ADDRESS>and  Nathan Shank  Moravian College 1200 Main
Street Bethlehem, PA 18018<EMAIL_ADDRESS>
(Date: (May 2021), and in revised form (July 2021))

###### Abstract.

In the $k$-diameter component connectivity model a network is consider
operational if there is a component with diameter at least $k$. Therefore, a
network is in a failure state if every component has diameter less than $k$.
In this paper we find the vertex variant of the $k$-diameter component
connectivity parameter, which is the minimum number of vertex deletions in
order to put a network into a failure state, for particular classes of graphs.
We also show the mixed variant by allowing vertex and edge failures within the
network. We show results for paths, cycles, complete, and complete bipartite
graphs for both variants as well as perfect $r$-ary trees for the vertex
variant.

###### Key words and phrases:

network reliability, connectivity, conditional connectivity, edge failure,
vertex connectivity, mixed connectivity, graph theory

###### 2010 Mathematics Subject Classification:

Primary. 94C15

##  1\. Introduction

Many different network structures can be modeled through graph theory. We
think of nodes, hubs, people, stations, objects, etc. as vertices of a graph
and the communication or connection between them as the edges of the graph.
For many reasons, accidental and deliberate, these networks break or fail.
Therefore, understanding the reliability and vulnerability of a network is
crucial to maintaining and building a reliable network .

When considering network reliability there are two issues to understand: what
are the minimum requirements to maintain an operational network and what
pieces of the network may fail. In a network modeled as a graph we can have
edges, vertices, or both (mixed) fail. To keep a network operational we often
consider what characterizes a failure state for a network. Several different
network reliability models have been studied and Harary [11] provided the
general framework for these network reliability models by considering a
property $P$ and a network, $G$. He defined a network to be operational if
there is a component of $G$ which has property $P$ and therefore, if no
component of $G$ has property $P$ the network is in a failure state.
Therefore, the network reliability of $G$ based on property $P$ is the minimum
number of failures so that no component of $G$ has property $P$.

For example, the component order edge connectivity [3] considers the minimum
number of edges that need to be removed from a graph so that all the
components have order less that some specific bound. Similarly the component
order vertex connectivity [5] considers vertex removal. For a survey of
results see [7].

In this paper we will consider a graph operational if it contains a component
of diameter at least $k$ for some fixed positive integer $k$. In practice, a
network may need to have a component with a minimum diameter for reliability
testing of a particular function, the spread of information must travel a
minimum distance before it is deemed valuable, or a virus which is transmitted
to neighbors may stay dormant until it has passed through a specific number of
hosts or nodes. These, and other examples, motivate the need to consider
networks which contain a component of minimum diameter. In [9], the authors
considered the instance where edges fail, or are removed. In this paper we
consider vertex failure as well as the mixed failure case for certain graph
classes.

##  2\. Background and Definitions

We will be using common graph theory notation found in [14]. Throughout we
will assume that $G=(V,E)$ is a finite simple graph with vertex set $V$ and
edge set $E$. For any edge set $D\subseteq E$, let $G-D$ denote the spanning
subgraph of $G$ containing the vertex set $V$ and the edge set $E-D$. For any
vertex set $H\subseteq V$, let $G-H$ denote the subgraph of $G$ induced by
$V-H$. Similarly, if $V^{\prime}\subseteq V$ and $E^{\prime}\subseteq
E(G-V^{\prime})$ we will write $G-V^{\prime}-E^{\prime}$ to denote
$(G-V^{\prime})-E^{\prime}$. For any set $A$, let $|A|$ denote the cardinality
of $A$.

If $u,v\in V$, let $d_{G}(u,v)$ denote the distance between $u$ and $v$ in $G$
(length of the shortest $u-v$ path in $G$). If the graph $G$ is clear, we will
denote $d_{G}(u,v)=d(u,v)$. If $d(u,v)=k$ for some positive integer $k$, then
we will say that $u$ is a k-neighbor of $v$, $\\{u,v\\}$ a $k-pair$, and a
$u-v$ path of length $k$ will be a $k$-path. If there exists a vertex $x\in V$
so that $d(u,x)=k$, then we will say that $u$ has a $k-neighbor$ in $G$. If
$G$ is a connected graph then the diameter of $G$ is the maximum distance
between any two vertices. If $G$ is not connected, the diameter is defined to
be infinite. A component of a graph $G$ is a connected and induced subgraph of
$G$, call it $H$, so that no other vertex in $v\in V(G-H)$ is adjacent to a
vertex in $H$. Clearly, if $G$ has more than one component, then the diameter
of $G$ is infinite.

Throughout we will consider $k$ to be a positive integer and we will consider
a network to be operational if there is a component of diameter at least $k$.
Thus, a network is in a failure state if every component has diameter less
than $k$. We can easily render a network into a failure state by removing all
of the vertices or all of the edges. However, we are interested in finding the
minimum number of vertex or edge deletions to produce a failure state.

Clearly, if $k\geq 2$ and if a graph contains a $k$-pair, then the graph also
contains a $(k-1)$-pair. This leads to the following lemma which will be
useful for our considerations of vertex and edge connectivity.

######  Lemma 2.1.

Let $G=(V,E)$ be a graph and $k$ be a positive integer. The graph $G$ is in a
failure state if and only if there does not exist a $k$-pair in $G$.

Thus, in the process of making a failure state we must remove vertices or
edges that impact each $k$-path. The following lemma shows that vertex
disjoint $k$-paths can not both be impacted by only one vertex or edge
removal. This will help us to show bounds on the number of vertex or edge
removals we need to render our network into a failure state.

######  Lemma 2.2.

Let $G=(V,E)$ be a graph and $k$ be a positive integer. If there exists $m$
vertex disjoint $k$-paths in $G$, then for any $v\in V$ or $e\in E$,
$G-\\{v\\}$ and $G-\\{e\\}$ each have at least $m-1$ vertex disjoint
$k$-paths.

###### Proof.

Let $G=(V,E)$ be a graph and $k$ be a positive integer. Assume $G$ has $m$
vertex disjoint $k$-paths. Assume by way of contradiction that there exists
some $v\in V$ such that $G-\\{v\\}$ has less than $m-1$ vertex disjoint
$k$-paths. Then $v$ was a vertex in at least two of the vertex disjoint
$k$-paths. Hence, $G$ did not contain $m$ vertex disjoint $k$-paths. A similar
argument holds for edge removals.

It is often the case that edges are the object that fails. The $k$-diameter
component edge connectivity of the graph was introduced in [9] and results
were shown for path, complete, and complete bipartite graphs as well as
perfect $r$-ary trees. In this paper we focus on the $k$-diameter component
vertex connectivity parameter as well as the mixed parameter which allows
vertex and edge deletions.

######  Definition 2.1.

Let $G=(V,E)$ be a graph and $k$ be a positive integer. A set
$V^{\prime}\subseteq V$ is a $k$-diameter component vertex disconnecting set
if $G-V^{\prime}$ has no vertex with a $k$-neighbor.

This means that a vertex set $V^{\prime}$ is a $k$-diameter component vertex
disconnecting set if every component of $G-V^{\prime}$ has diameter less than
$k$. If $V^{\prime}$ is a $k$-diameter component vertex disconnecting set,
then $G-V^{\prime}$ is in a failure state.

Recall we are interested in finding the minimum number of vertices that can be
removed to produce a failure state. This motivates the definition of the
$k$-diameter component vertex connectivity parameter.

######  Definition 2.2.

Given a graph $G=(V,E)$ and a positive integer $k$, the $k$-diameter component
vertex connectivity parameter of $G$, denoted $CV_{k}(G)$, is the size of the
smallest $k$-diameter component vertex disconnecting set.

Thus, the $k$-diameter component vertex connectivity parameter is the size of
the smallest vertex set $V^{\prime}$ so that $G-V^{\prime}$ is in a failure
state.

Similarly, we will consider edge disconnecting sets which will be used in our
mixed deletion case.

######  Definition 2.3.

Let $G=(V,E)$ be a graph and $k$ be a positive integer. A set
$E^{\prime}\subseteq E$ is a $k$-diameter component edge disconnecting set if
$G-E^{\prime}$ has no vertex with a $k$-neighbor.

This means that an edge set $E^{\prime}$ is a $k$-diameter component edge
disconnecting set if every component of $G-E^{\prime}$ has diameter less than
$k$. If $E^{\prime}$ is a$k$-diameter component edge disconnecting set, then
$G-E^{\prime}$ is in a failure state.

As with vertex deletions, we can also consider the minim number of edges whose
removal produces a failure state. This motivates the definition of the
$k$-diameter component edge connectivity parameter.

######  Definition 2.4.

Given a graph $G=(V,E)$ and a positive integer $k$, the $k$-diameter component
edge connectivity parameter of $G$, denoted $CE_{k}(G)$, is the size of the
smallest $k$-diameter component edge disconnecting set.

Thus, the $k$-diameter component edge connectivity parameter is the minimum
size of an edge set $E^{\prime}$ so that $G-E^{\prime}$ is in a failure state.

It is often the case that vertices and edges fail which is investigated
through the mixed deletion case. This was first introduced by Beineke and
Harary [1]. The following definitions address the $k$-diameter component
connectivity function, which is a mixed version of the $k$-diameter component
connectivity involving both vertex and edge deletions. As is standard we will
remove vertices first then remove edges.

######  Definition 2.5.

Let $G=(V,E)$ be a graph, $k$ be a positive integer, and
$p\in\\{0,1,...,CV_{k}(G)\\}$. Then the $k$-diameter component connectivity
function of $G$ is defined as
$CM_{k}(G,p)=\text{min}\\{CE_{k}(G-V^{\prime}):V^{\prime}\subseteq
V,|V^{\prime}|=p\\}$.

So $CM_{k}(G,p)$ is the minimum number of edges that must be removed to render
the graph into a failure state assuming we can also remove any $p$ vertices in
the graph. Note that we must remove the $p$ vertices first then remove the
least amount of edges.

######  Definition 2.6.

Let $G=(V,E)$ be a graph and $k$ be a positive integer. A $k$-diameter
component connectivity pair of $G$ for each $p\in\\{0,1,...,CV_{k}(G)\\}$ is
an ordered pair $(p,q)$, such that $CM_{k}(G,p)=q$.

Two obvious connectivity pairs of $G$ are $(0,CE_{k}(G))$ and $(CV_{k}(G),0)$.
For each value of $p$ where $0\leq p\leq CV_{k}(G)$, there is a unique
$k$-diameter component connectivity pair.

When trying to find the value of $CM_{k}(G,p)$ it is often useful to consider
which $p$ vertices we need to remove to minimize $CE_{k}(G-V^{\prime})$. This
motivates the following definition of a optimal $p$-set.

######  Definition 2.7.

Let $G=(V,E)$ be a graph, $k$ be a positive integer, and $p$ be a nonnegative
integer. Let $V^{\prime}\subseteq V$ such that $|V^{\prime}|=p$. We say
$V^{\prime}$ is an optimal $p$-set if $CM_{k}(G,p)=CE_{k}(G-V^{\prime})$.

The following lemma will prove valuable for providing lower bounds. The lemma
shows disjoint $k$-paths provide a lower bound for the number of vertex
deletions, edge deletions, or mixed deletions needed to produce a failure
state.

######  Lemma 2.3.

Let $G=(V,E)$ be a graph and let $k$ be a positive integer. If there exists
$M$ vertex disjoint $k$-paths in $G$, then $CV_{k}(G)\geq M$ and
$CE_{k}(G)\geq M$. Furthermore, if $CM_{k}(G,p)=q$, then $p+q\geq M$.

###### Proof.

Let $G=(V,E)$ be a graph and let $k$ be a positive integer. Assume there
exists $M$ vertex disjoint $k$-paths in $G$. Let $V^{\prime}\subseteq V$ such
that $G-V^{\prime}$ is in a failure state. Let $E^{\prime}\subseteq E$ such
that $G-E^{\prime}$ is in a failure state. Then Lemma 2.1 and multiple
iterations of Lemma 2.2 implies that $|V^{\prime}|\geq M$ and
$|E^{\prime}|\geq M$.

Let $V^{*}\subseteq V$ and $E^{*}\subseteq E(G-V^{*})$ such that
$G-V^{*}-E^{*}$ is in a failure state. Then Lemma 2.1 and Lemma 2.2 implies
that $|V^{*}|+|E^{*}|\geq M$. Hence, if $CM_{k}(G,p)=q$, then $p+q\geq M$.

##  3\. Vertex Deletion Results

In this section we will consider only vertex deletions; we compute $CV_{k}(G)$
for specific graphs $G$. We provide results for path graphs, cycles, complete
graphs, complete bipartite graphs, and perfect $r$-ary trees. Note that if
$k=1$, then $CV_{k}$ is the minimum number of vertex deletions whose removal
results in an edgeless graph. Therefore, we will always assume $k\geq 2$.

###  3.1. Path Graphs

Consider the path graph on $n$ vertices, denoted $P_{n}$. Label the vertices
consecutively from 1 to $n$ starting at a pendant vertex. Since any path of
length $k$ has $k+1$ vertices, there are
$\left\lfloor\frac{n}{k+1}\right\rfloor$ vertex disjoint $k$-paths in $P_{n}$.
By Lemma 2.3, $CV_{k}(P_{n})\geq\left\lfloor\frac{n}{k+1}\right\rfloor$.

If we delete every vertex whose label is a multiple of $k+1$, then all of the
remaining components have $k$ vertices, with the exception of at most one
component which could have fewer than $k$ vertices. Therefore, the diameter of
each remaining component will be less than $k$. This results in a total of
$\left\lfloor\frac{n}{k+1}\right\rfloor$ deletions. Hence,
$CV_{k}(P_{n})\leq\left\lfloor\frac{n}{k+1}\right\rfloor$. These two
observations imply the following:

######  Theorem 3.1.

For every positive integer n,

$CV_{k}(P_{n})=\left\lfloor\frac{n}{k+1}\right\rfloor.$

###  3.2. Cycle Graphs

Consider the cycle graph on $n$ vertices, denoted $C_{n}$. Since
$diam(C_{n})=\left\lfloor\frac{n}{2}\right\rfloor$, if
$k>\left\lfloor\frac{n}{2}\right\rfloor$, then $C_{n}$ is already in a failure
state and no deletions are necessary. If
$k\leq\left\lfloor\frac{n}{2}\right\rfloor$, then at least one deletion must
be made. Notice that the deletion of any single vertex from $C_{n}$ leaves a
path graph on $n-1$ vertices. Then, by Theorem 3.1,
$CV_{k}(C_{n})=\left\lfloor\frac{n-1}{k+1}\right\rfloor+1$. Hence, we have the
following:

######  Theorem 3.2.

For every positive integer n,

$CV_{k}(C_{n})=\begin{cases}0&\text{if
}k>\left\lfloor\frac{n}{2}\right\rfloor\\\
\left\lfloor\frac{n+k}{k+1}\right\rfloor&\text{if
}k\leq\left\lfloor\frac{n}{2}\right\rfloor.\\\ \end{cases}$

###  3.3. Complete Graphs

Consider the complete graph on $n$ vertices, denoted $K_{n}$. Since the
diameter of a complete graph is 1 and $k\geq 2$, $K_{n}$ is already in a
failure state. Thus, we see the following obvious result:

######  Theorem 3.3.

For any positive integer $n$,

$CV_{k}(K_{n})=0.$

###  3.4. Complete Bipartite Graph

Now we will consider a complete bipartite graph $K_{a,b}=(V,E)$ with parts $A$
and $B$ where $V=A\cup B$, $A\cap B=\emptyset$, $|A|=a>0$, and $|B|=b>0$.

######  Theorem 3.4.

For any positive integer $a$ and $b$,

$CV_{k}(K_{a,b})=\begin{cases}0&\text{if }a=b=1,\\\ 0&\text{if }k>2\text{ and
}\max\\{a,b\\}\geq 2,\\\ \text{min}\\{a,b\\}&\text{if }k=2\text{ and
}\max\\{a,b\\}\geq 2.\\\ \end{cases}$

###### Proof.

Let $K_{a,b}=(V,E)$ be a complete bipartite graph with parts $A$ and $B$ where
$V=A\cup B$, $A\cap B=\emptyset$, $|A|=a>0$, and $|B|=b>0$. The diameter of a
complete bipartite graph is $2$ unless $a=b=1$, in which case the diameter is
1 and $K_{1,1}$ is already in a failure state for all $k>1$. Consider when
$a\geq 2$ or $b\geq 2$. If $k>2$, then $K_{a,b}$ is already in a failure
state. Now consider when $k=2$. The only induced subgraphs of $K_{a,b}$ which
are in a failure state are $K_{1,1}$, subgraphs of $A$, and subgraphs of $B$.
To produce $K_{1,1}$, we must delete all but two vertices: one vertex from $A$
and one vertex from $B$. Thus, the resulting number of vertex deletions is
$(a-1)+(b-1)$. Since $A$ is the subgraph of $A$ with the most vertices, we
only need to consider deleting vertices to produce $A$. In order to produce
$A$, we must delete all vertices from $B$ and, therefore, the resulting number
of vertex deletions is $b$. Similarly, to produce $B$, $a$ vertex deletions
are necessary. If either $a\geq 2$ or $b\geq 2$, then it is easily seen that
$(a-1)+(b-1)\geq\text{min}\\{a,b\\}$.

###  3.5. Perfect $r$-ary Trees

Throughout we will assume $r$ and $l$ are positive integers and let
$T_{r,l}=(V,E)$ be a perfect $r-ary$ tree with height $l$. This means that
$T_{r,l}$ has $\frac{r^{l+1}-1}{r-1}$ vertices and
$\frac{r(r^{l+1}-1)}{r-1}-r$ edges. We can enumerate the vertices and edges of
$T_{r,l}$ as follows:

$\displaystyle V$ $\displaystyle=\\{v_{i,j}:1\leq i\leq l+1,1\leq j\leq
r^{(l+1)-i}\\},\text{ and }$ $\displaystyle E$
$\displaystyle=\\{(v_{i,j},v_{i-1,m}):2\leq i\leq l+1,1\leq j\leq
r^{(l+1)-I},$ $\displaystyle\indent\indent(j-1)r+1\leq m\leq jr\\}.$

We will say that vertex $v_{i,j}\in V$ is on level $i$. Notice that the root
vertex is on level $l+1$ and the leaves are on level $1$.

Figure 1. A complete 2-ary tree with height 3 ($r=2,l=3$)

Let $h$ be a positive integer. Fix a vertex $v_{i,j}\in V$, with $i>h$ and
consider the subtree $T_{v_{i,j}}^{h}$ of $T_{r,l}$ induced by $v_{i,j}$ and
all of its descendants at a distance at most $h$. Notice in $T_{v_{i,j}}^{h}$,
the degree of $v_{i,j}$ is $r$ and any vertex $x$ of degree 1 in
$T_{v_{i,j}}^{h}$ will satisfy $d(x,v_{i,j})=h$. Also notice that
$T_{v_{i,j}}^{h}$ is a perfect $r$-ary tree of height $h$.

The following lemma establishes a set $V^{\prime}$ of vertices which will form
our minimum $k$-diameter component vertex disconnecting set for $T_{r,l}$. The
cardinality of this set is shown so that we can prove it is in fact the
minimum $k$-diameter component vertex disconnecting set.

######  Lemma 3.1.

Let $T_{r,l}=(V,E)$. Let $V^{\prime}\subset V$ such that

$V^{\prime}=\left\\{v_{m\left(\left\lceil\frac{k}{2}\right\rceil+1\right),j}\in
V:1\leq
m\leq\left\lfloor\frac{l+1}{\left\lceil\frac{k}{2}\right\rceil+1}\right\rfloor,1\leq
j\leq r^{l+1-m\left(\left\lceil\frac{k}{2}\right\rceil+1\right)}\right\\}.$

Then,

$|V^{\prime}|=\frac{r^{l+1}-r^{l+1-\left\lfloor\frac{l+1}{\left\lceil\frac{k}{2}\right\rceil+1}\right\rfloor\left(\left\lceil\frac{k}{2}\right\rceil+1\right)}}{r^{\left\lceil\frac{k}{2}\right\rceil+1}-1}.$

###### Proof.

Consider $V^{\prime}$ as defined above. Then by summing over all possible
choices of $m$ we see

$\displaystyle|V^{\prime}|=\sum_{m=1}^{\left\lfloor\frac{l+1}{\left\lceil\frac{k}{2}\right\rceil+1}\right\rfloor}r^{l+1-m\left(\left\lceil\frac{k}{2}\right\rceil+1\right)}.$

Simplifying the previous expression, we see:

$\displaystyle|V^{\prime}|$
$\displaystyle=r^{l+1}\sum_{m=1}^{\left\lfloor\frac{l+1}{\left\lceil\frac{k}{2}\right\rceil+1}\right\rfloor}r^{-m\left(\left\lceil\frac{k}{2}\right\rceil+1\right)}$
$\displaystyle=r^{l+1}\left(\frac{1-r^{-\left\lfloor\frac{l+1}{\left\lceil\frac{k}{2}\right\rceil+1}\right\rfloor\left(\left\lceil\frac{k}{2}\right\rceil+1\right)}}{r^{\left\lceil\frac{k}{2}\right\rceil+1}-1}\right)$
$\displaystyle=\frac{r^{l+1}-r^{l+1-\left\lfloor\frac{l+1}{\left\lceil\frac{k}{2}\right\rceil+1}\right\rfloor\left(\left\lceil\frac{k}{2}\right\rceil+1\right)}}{r^{\left\lceil\frac{k}{2}\right\rceil+1}-1}.$

Now we will show $CV_{k}(T_{r,l})=|V^{\prime}|$. For each $v_{i,j}\in
V^{\prime}$, $T_{v_{i,j}}^{\left\lceil\frac{k}{2}\right\rceil}$ is not in a
failure state. We will also show that for any
$v_{i,j},v_{i^{\prime},j^{\prime}}\in V^{\prime}$ with $v_{i,j}\neq
v_{i^{\prime},j^{\prime}}$, $T_{v_{i,j}}^{\left\lceil\frac{k}{2}\right\rceil}$
and $T_{v_{i^{\prime},j^{\prime}}}^{\left\lceil\frac{k}{2}\right\rceil}$ are
disjoint. Thus, for each vertex in $V^{\prime}$, we need at least one vertex
deletion to produce a subgraph of $T_{r,l}$ which is in a failure state. We
will also show that $T_{r,l}-V^{\prime}$ is in a failure state and, therefore,
$V^{\prime}$ forms a minimum $k$-diameter disconnecting set. For the sake of
simplicity, we will denote $T_{v_{i,j}}^{\left\lceil\frac{k}{2}\right\rceil}$
with $T_{v_{i,j}}$.

######  Theorem 3.5.

Let $r,l$ and $k$ be positive integers. Let $T_{r,l}=(V,E)$ be a perfect
$r-ary$ tree with height $l$. Then,

$CV_{k}(T_{r,l})=\frac{r^{l+1}-r^{l+1-\left\lfloor\frac{l+1}{\left\lceil\frac{k}{2}\right\rceil+1}\right\rfloor\left(\left\lceil\frac{k}{2}\right\rceil+1\right)}}{r^{\left\lceil\frac{k}{2}\right\rceil+1}-1}.$

###### Proof.

Let $r,l$, and $k$ be positive integers. Let $T_{r,l}=(V,E)$.

First we will establish that

$CV_{k}(T_{r,l})\geq\frac{r^{l+1}-r^{l+1-\left\lfloor\frac{l+1}{\left\lceil\frac{k}{2}\right\rceil+1}\right\rfloor\left(\left\lceil\frac{k}{2}\right\rceil+1\right)}}{r^{\left\lceil\frac{k}{2}\right\rceil+1}-1}$

by finding a set of disjoint $k$-pairs. Let

$V^{\prime}=\left\\{v_{m\left(\left\lceil\frac{k}{2}\right\rceil+1\right),j}\in
V:1\leq
m\leq\left\lfloor\frac{l+1}{\left\lceil\frac{k}{2}\right\rceil+1}\right\rfloor,1\leq
j\leq r^{l+1-m\left(\left\lceil\frac{k}{2}\right\rceil+1\right)}\right\\}.$

For each $v_{i,j}\in V^{\prime}$ let $T_{v_{i,j}}$ be the subgraph induced on
$v_{i,j}$ and all of its descendants at a distance at most
$\left\lceil\frac{k}{2}\right\rceil$. Notice
$i>\left\lceil\frac{k}{2}\right\rceil$ for each $v_{i,j}\in V^{\prime}$ and
$T_{v_{i,j}}$ is a perfect $r$-ary tree with height
$\left\lceil\frac{k}{2}\right\rceil$ for all $v_{i,j}\in V^{\prime}$. Since
the diameter of a perfect $r$-ary tree of height $a$ is $2a$, we have
$diam(T_{v_{i,j}})=2\left\lceil\frac{k}{2}\right\rceil$ for each $v_{i,j}\in
V^{\prime}$. This implies, since $k$ is an integer,

$k+1\geq diam(T_{v_{i,j}})\geq k.$

Since $diam(T_{v_{i,j}})\geq k$, $T_{v_{i,j}}$ contains at least one $k$-pair
and, thus, is not in a failure state.

Let $v_{i,j},v_{i,j^{\prime}}\in V^{\prime}$ with $j\neq j^{\prime}$. We will
show that all $T_{v_{i,j}}$ and $T_{v_{i,j^{\prime}}}$ are disjoint. Since
$v_{i,j}$ and $v_{i,j^{\prime}}$ have a common ancestor, they cannot share any
decendants, or $T_{r,l}$ would contain a cycle and is not a tree. Therefore,
$T_{v_{i,j}}$ and $T_{v_{i,j^{\prime}}}$ are disjoint.

Consider $v_{i,j},v_{i^{\prime},j^{\prime}}\in V^{\prime}$ where $i\neq
i^{\prime}$. We will show that $T_{v_{i,j}}$ and
$T_{v_{i^{\prime},j^{\prime}}}$ are disjoint. By the definition of
$V^{\prime}$, vertices on different levels in $V^{\prime}$ are at a distance
of at least $\left\lceil\frac{k}{2}\right\rceil+1$ from each other, so
$v_{i,j}\not\in T_{v_{i^{\prime},j^{\prime}}}$ and
$v_{i^{\prime},j^{\prime}}\not\in T_{v_{i,j}}$. Since $v_{i,j}$ and
$v_{i^{\prime},j^{\prime}}$ are the roots of $T_{v_{i,j}}$ and
$T_{v_{i^{\prime},j^{\prime}}}$ respectively, and $T_{v_{i,j}}$ and
$T_{v_{i^{\prime},j^{\prime}}}$ are trees of height
$\left\lceil\frac{k}{2}\right\rceil$, this implies $T_{v_{i,j}}$ and
$T_{v_{i,j^{\prime}}}$ are disjoint.

Therefore, we have shown that $\\{T_{v_{i,j}}:v_{i,j}\in V^{\prime}\\}$ are
pairwise disjoint.

Since $T_{v_{i,j}}$ contains a $k$-pair for each $v_{i,j}\in V^{\prime}$,
there are at least $|V^{\prime}|$ disjoint $k$-pairs in $T_{r,l}$. Therefore,
by Corollary 2.3 and Lemma 3.1,

$CV_{k}(T_{r,l})\geq\frac{r^{l+1}-r^{l+1-\left\lfloor\frac{l+1}{\left\lceil\frac{k}{2}\right\rceil+1}\right\rfloor(\left\lceil\frac{k}{2}\right\rceil+1)}}{r^{\left\lceil\frac{k}{2}\right\rceil+1}-1}.$

Now we will show that

$CV_{k}(T_{r,l})\leq\frac{r^{l+1}-r^{l+1-\left\lfloor\frac{l+1}{\left\lceil\frac{k}{2}\right\rceil+1}\right\rfloor\left(\left\lceil\frac{k}{2}\right\rceil+1\right)}}{r^{\left\lceil\frac{k}{2}\right\rceil+1}-1}$

by showing $T_{r,l}-V^{\prime}$ is in a failure state.

Consider $T_{r,l}-V^{\prime}$. By deleting all vertices in $V^{\prime}$ from
$T_{r,l}$, we are deleting entire levels of vertices. In fact, we are deleting
all vertices which are on a level which is an integer multiple of
$\left\lceil\frac{k}{2}\right\rceil+1$. Notice, then, that
$T_{r,l}-V^{\prime}$ is a disconnected graph where each component is a perfect
$r$-ary tree. All of these trees have height
$\left\lceil\frac{k}{2}\right\rceil-1$, except the tree containing the root
vertex, which could have less. Then,
$diam(T_{r,\left\lceil\frac{k}{2}\right\rceil-1})=2(\left\lceil\frac{k}{2}\right\rceil-1)\leq
2(\frac{k+1}{2})-2\leq k-1$. Therefore, each component has diameter less than
$k$, and $G-V^{\prime}$ is in a failure state. Hence,

$CV_{k}(T_{r,l})\leq\frac{r^{l+1}-r^{l+1-\left\lfloor\frac{l+1}{\left\lceil\frac{k}{2}\right\rceil+1}\right\rfloor\left(\left\lceil\frac{k}{2}\right\rceil+1\right)}}{r^{\left\lceil\frac{k}{2}\right\rceil+1}-1}.$

Combining these two inequalities, we see that

$CV_{k}(T_{r,l})=\frac{r^{l+1}-r^{l+1-\left\lfloor\frac{l+1}{\left\lceil\frac{k}{2}\right\rceil+1}\right\rfloor\left(\left\lceil\frac{k}{2}\right\rceil+1\right)}}{r^{\left\lceil\frac{k}{2}\right\rceil+1}-1}.$

##  4\. Mixed Deletion Results

Now we will investigate the $k$-diameter component connectivity function of a
few simple graph classes. We provide results for path graphs, cycles, complete
graphs, and complete bipartite graphs. As with the previous section, we will
assume throughout that $k\geq 2$ is a positive integer.

###  4.1. Path Graphs

To decompose path graphs into failure states, we will create path graphs
components of maximum length which are in a failure state.

######  Theorem 4.1.

Let $P_{n}$ be the path on $n$ vertices. For any nonnegative integer $p\leq
CV_{k}(P_{n})$,

$CM_{k}(P_{n},p)=\begin{cases}0&\text{if
}p=\left\lfloor\frac{n}{k+1}\right\rfloor\\\ \\\
\left\lfloor\frac{n-p(k+1)-1}{k}\right\rfloor&\text{if
}p<\left\lfloor\frac{n}{k+1}\right\rfloor.\\\ \end{cases}$

###### Proof.

Let $n$ be a positive integer. Consider $P_{n}$ and let $p\leq CV_{k}(P_{n})$
be a nonnegative integer. If $p=\left\lfloor\frac{n}{k+1}\right\rfloor$, then
by Theorem 3.1, $CM_{k}(P_{n},p)=0$.

Assume $p<\left\lfloor\frac{n}{k+1}\right\rfloor$. Label vertices of $P_{n}$
consecutively starting at a pendant vertex so that
$V=\\{v_{i}:i\in\mathbb{Z},1\leq i\leq n\\}$. Let $V^{\prime}\subset V$ such
that $V^{\prime}=\\{v_{j(k+1)}:j\in\mathbb{Z},1\leq j\leq p\\}$. Then
$P_{n}-V^{\prime}$ has $p$ components with diameter $k-1$ and one component,
denoted $C$, which is a path on $n-p(k+1)$ vertices. By Theorem 3.1 in [9],
$CE_{k}(C)=\left\lfloor\frac{n-p(k+1)-1}{k}\right\rfloor$. Let $E^{\prime}$ be
the set of $\left\lfloor\frac{n-p(k+1)-1}{k}\right\rfloor$ edges removed from
component $C$ to produce a failure state. Then $P_{n}-V^{\prime}-E^{\prime}$
is in a failure state. Therefore,
$CM_{k}(P_{n},p)\leq\left\lfloor\frac{n-p(k+1)-1}{k}\right\rfloor$.

Assume we are going to delete a set, $P$, of vertices where $|P|=p$ and a set,
$Q$, of edges where $|Q|=q$ so that $P_{n}-P-Q$ is in a failure state. Since
$P_{n}$ is a tree, any edge or vertex deletion creates at most one new
component. Therefore, $P_{n}-P-Q$ can have at most $p+q+1$ components. If
$P_{n}-P-Q$ is in a failure state, then each component contains at most $k$
vertices. Thus, the number of vertices in our original graph is the sum of all
the vertices in the failed components plus the $p$ vertices we had to delete
from $P$. Therefore, the number of vertices in our original graph would be
bounded above by $(p+q+1)k+p.$ We will now use this fact to show that
$CM_{k}(P_{n},p)=\left\lfloor\frac{n-p(k+1)-1}{k}\right\rfloor$.

Assume by way of contradiction that there exists some set $V^{*}\subseteq V$
with $|V^{*}|=p$ and $E^{*}\subseteq E$ with
$|E^{*}|=\left\lfloor\frac{n-p(k+1)-1}{k}\right\rfloor-1$ such that
$P_{n}-V^{*}-E^{*}$ is in a failure state. As shown above, if
$P_{n}-V^{*}-E^{*}$ is in a failure state, then the number of vertices in
$P_{n}$ is bounded above by

$|V(P_{n})|\leq\left(p+\left(\left\lfloor\frac{n-p(k+1)-1}{k}\right\rfloor-1\right)+1\right)k+p.$

Simplifying this expression, we see

$\displaystyle|V(P_{n})|$ $\displaystyle\leq
p(k+1)+k\left\lfloor\frac{n-p(k+1)-1}{k}\right\rfloor$ $\displaystyle\leq
p(k+1)+k\left(\frac{n-p(k+1)-1}{k}\right)$ $\displaystyle=n-1.$

Hence, we have a contradiction, because $|V(P_{n})|=n$.

###  4.2. Cycle Graphs

A cycle is only one vertex deletion away from becoming a path graph.
Therefore, we can use our results for path graphs to analyze cycle graphs.

######  Theorem 4.2.

Consider $C_{n}$, the cycle on $n$ vertices. For any positive integer $n$ and
nonnegative integer $p\leq CV_{k}(C_{n})$,

$CM_{k}(C_{n},p)=\begin{cases}0&\text{if
}k>\left\lfloor\frac{n}{2}\right\rfloor\\\ 0&\text{if
}k\leq\left\lfloor\frac{n}{2}\right\rfloor\text{ and
}p=\left\lfloor\frac{n+k}{k+1}\right\rfloor\\\
\left\lfloor\frac{n-p(k+1)-1}{k}\right\rfloor+1&\text{if
}k\leq\left\lfloor\frac{n}{2}\right\rfloor\text{and
}0<p<\left\lfloor\frac{n+k}{k+1}\right\rfloor.\\\ \end{cases}$

###### Proof.

Let $C_{n}$ be the cycle graph on $n$ vertices. Note that
$diam(C_{n})=\left\lfloor\frac{n}{2}\right\rfloor$, so if
$k>\left\lfloor\frac{n}{2}\right\rfloor$, $C_{n}$ is already in a failure
state and requires no deletions. Consider when
$p=\left\lfloor\frac{n+k}{k+1}\right\rfloor$. Then by Theorem 3.2, there
exists a set $V^{\prime}$ of $p$ vertices such that $C_{n}-V^{\prime}$ is in a
failure state. Hence, if $p=\left\lfloor\frac{n+k}{k+1}\right\rfloor$, then
$CM_{k}(C_{n},p)=0$.

Consider when $k\leq\left\lfloor\frac{n}{2}\right\rfloor$ and
$0<p<\left\lfloor\frac{n+k}{k+1}\right\rfloor$. Note that any vertex deletion
leaves a path on $n-1$ vertices, hence, $CM_{k}(C_{n},p)=CM_{k}(P_{n-1},p-1)$.
Therefore, by Theorem 4.1
$CM_{k}(C_{n},p)=\left\lfloor\frac{(n-1)-(p-1)(k+1)-1}{k}\right\rfloor$.
Simplifying this expression, we see

$CM_{k}(C_{n},p)=\left\lfloor\frac{n-p(k+1)-1}{k}\right\rfloor+1.$

###  4.3. Complete Graphs

Consider the complete graph on $n$ vertices, denoted $K_{n}$. Since any
complete graph has diameter 1, we see $K_{n}$ is already in a failure state
since we are assuming $k\geq 2$. Thus, we see the following obvious result:

######  Theorem 4.3.

For every positive integer $n$,

$CM_{k}(K_{n},p)=0.$

###  4.4. Complete Bipartite Graphs

######  Theorem 4.4.

Consider the complete bipartite graph $K_{(a,b)}$ where $a$ and $b$ are
positive integers with $a\leq b$. Then for any nonnegative integer $p\leq
CV_{k}(K_{a,b})$,

$CM_{k}(K_{a,b},p)=\begin{cases}0&\text{if }a=b=1\\\ 0&\text{if }k>2\text{ and
}b>1\\\ (a-p)(b-1)&\text{if }k=2\text{ and }b>1.\\\ \end{cases}$

###### Proof.

Let $k$ be a positive integer. Consider the complete bipartite graph, denoted
$K_{a,b}=(V,E)$ with parts $A$ and $B$ where $V=A\cup B$, $A\cap B=\emptyset$,
$|A|=a>0$, and $|B|=b>0$. Furthermore, assume without loss of generality that
$a\leq b$.

If $a=b=1$, then $diam(K_{a,b})=1$ and $K_{a,b}$ is in a failure state. If
$b\geq 2$, then $diam(K_{a,b})=2$. If $k>2$, then $K_{a,b}$ is in a failure
state. If $k=2$, then $K_{a,b}$ is not in the failure state. Therefore, fix
$k=2$ for the remainder of the proof.

Note that min$\\{a,b\\}=a$, so Theorem 3.4 implies that $CM_{k}(K_{a,b},a)=0$.
Thus, assume $p<a$. Let $V^{\prime}$ be a set of vertices deleted from
$K_{a,b}$ such that $|V^{\prime}|=p$. We claim that $V^{\prime}\subseteq A$ is
an optimal $p$-set. In other words, it is optimal to delete all $p$ vertices
from part $A$ of $K_{a,b}$. Then by, Theorem 3.3 of [9],
$CM_{k}(K_{(a,b)})=CE_{k}(K_{a-p,b})=(a-p)(b-1)$.

Assume by way of contradiction that $V^{\prime}\subseteq A$ is not an optimal
$p$-set. Then there exists an optimal $p$-set $V^{*}=V_{A}\cup V_{B}$, where
$V_{A}\subseteq A$, $V_{B}\subseteq B$, $|V_{A}|=x$, $|V_{B}|=y$, $y\geq 1$
and $x+y=p$. In other words, it is optimal to delete some vertices from part
$A$ and some from part $B$ of $K_{a,b}$. Then, we have the following two
cases.

Case 1: Assume $a-x\leq b-y$. Then if $V^{*}$ is an optimal $p$-set, by
Theorem 3.3 of [9], $CM_{k}(K_{a,b},p)=CE_{k}(K_{a-x,b-y})=(a-x)(b-y-1)$.

Note that $y\geq 1$ implies

$y(a-x)\leq y(b-1).$

Then, by substituting $y=p-x$ on the right side of this inequality, we see

$x(b-1)+y(a-x)\leq p(b-1).$

Adding $a(b-1)$ to both sides of the above inequality and simplifying, we see

$(a-p)(b-1)\leq(a-x)(b-y-1).$

Hence, $CE_{k}(K_{a-p,b})\leq CE_{k}(K_{a-x,b-y})$.

Case 2: Assume $b-y<a-x$. Then if $V^{*}$ is an optimal $p$-set, by Theorem
3.3 of [9], $CM_{k}(K_{a,b},p)=CE_{k}(K_{a-x,b-y})=(b-y)(a-x-1)$.

Note that $a\leq b$ implies that $a-x\leq b$. Multiplying by $(y-1)$ and then
adding $b(a-x)$ to both sides of this inequality, we see

$(a-x)b+y(a-x)-(a-x)\leq(a-x)b+y(b-1)-b+y.$

Simplifying this expression, we see

$(a-x-y)(b-1)\leq(b-y)(a-x-1).$

Then substituting $p=x+y$, we see

$(a-p)(b-1)\leq(b-y)(a-x-1).$

Hence, $CE_{k}(K_{a-p,b})\leq CE_{k}(K_{a-x,b-y})$.

In either of these two cases, $CE_{k}(K_{a-p,b})=(a-p)(b-1)\leq
CE_{k}(K_{a-x,b-y})$. Hence, $V^{\prime}$ is an optimal $p$-set and

$CM_{k}(K_{a,b},p)=\begin{cases}0&\text{if }a=b=1\\\ 0&\text{if }k>2\text{ and
}b>1\\\ (a-p)(b-1)&\text{if }k=2\text{ and }b>1.\\\ \end{cases}$

## References

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11institutetext: Konkoly Observatory, Research Center for Astronomy and Earth
Sciences, Eötvös Loránd Research Network
Budapest, 1121 Konkoly Thege ut. 15-17, Hungary
11email<EMAIL_ADDRESS>
# Toward more accurate RR Lyrae metallicities

Geza Kovacs and Johanna Jurcsik

(Received 09-05-2023 / Accepted DD-MM-2023)

By using a large sample of published spectroscopic iron abundances, we point
out the importance of gravity correction in deriving more accurate metal
abundances for RR Lyrae stars. For the $197$ stars with multiple spectra we
find overall [Fe/H] standard deviations of $0.167$ (as published), $0.145$
(shifted by data source zero points) and $0.121$ (both zero point shifted and
gravity-corrected). These improvements are significant at the $\sim 2\sigma$
level at each correction step, leading to a clearly significant improvement
after both corrections applied. The higher quality of the gravity-corrected
metallicities is strongly supported also by the tighter correlation with the
metallicities predicted from the period and Fourier phase $\varphi_{31}$. This
work underlines the need for using some external estimates of the temporal
gravity in the chemical abundance analysis rather than relying on a full-
fetched spectrum fit that leads to large correlated errors in the estimated
parameters.

###### Key Words.:

Stars: abundances – stars: variables: RR Lyrae – stars: horizontal-branch

##  1 Introduction

Heavy elements – primarily [Fe/H] but also $\alpha$-elements – play a leading
role in the evolution of RR Lyrae stars and also in their applicability as
distance indicators and Galactic structure tracers. Unfortunately, they also
have highly variable envelope/atmosphere, that makes the applicability of
standard spectroscopic methods for chemical abundance analysis rather
sensitive to the pulsation phase at the moment of the observation.

In general, spectroscopic abundances are determined by direct model (or
spectral library/template) fits, by adjusting basic atmospheric parameters,
such as effective temperature $T_{eff}$, temporal gravity $\log g$ and
turbulent velocity $V_{t}$, together with the assumed element distribution
(see, e.g. Jofré et al., 2019, for an overview of the methodologies used to
produce today’s “industrial” stellar abundances). Unfortunately, this process
is rather ill-conditioned, and the parameters fitted are both correlated and
erroneous. To improve the conditioning, when possible, some of the parameters
(usually $T_{eff}$ and $\log g$) are determined externally, by using some
reliable parameter source (e.g., spectral energy density fit from multi-
waveband observations to derive $T_{eff}$). The gravity parameter is usually
the most difficult parameter to estimate. However, extrasolar planet host
stars (or the primary components of binaries with low-mass secondary
components) and pulsating variables are excellent candidates for giving good
estimates on the temporal value of $\log g$. For extrasolar planet host stars
the gravity is well estimated from the combination of the orbital and stellar
evolution model analyses (Noyes et al., 2008). For radially pulsating stars
the temporal gravity is very closely estimated from the period (e.g., Gough et
al., 1965; Dékány et al., 2008) and from the radial velocity curve (e.g.,
Clementini et al., 1995).

Considering only RRab (fundamental mode) variables, here we examine the effect
of employing a simple post-correction method on the already determined
spectroscopic metallicities to improve the accuracy of the final abundances.
In addition to the improvement of the internal accuracy, external relations
are also used to test the quality of the derived abundances.

Reference to the Appendices for some additional details of the analysis is
implicit throughout the paper. Extended materials related to this paper are
deposited at the CDS site.111Metallicities and light curve parameters used in
this paper are available at the CDS via http://cdsarc.u-strasbg.fr/viz-
bin/qcat?J/A+A/3digitVol/Apagenumber

##  2 Method

The main issue in the derivation of stellar parameters on purely spectroscopic
basis is that, in general, these parameters are difficult to disentangle. The
success of the process depends on many factors and the verification of the
reliability of the parameters derived are often presumptional (e.g., cluster’s
chemical homogeneity). For dynamical atmospheres the situation is coupled with
changing stellar atmospheric conditions and physical assumptions such as the
validity of local thermodinamical equilibrium.

Here we focus on the single problem of temporal gravity. The generally
employed spectroscopic element determination methods make this parameter free-
floating (together with the iron abundance and $T_{eff}$ – see, e.g. Crestani
et al., 2021). However, for classical radial pulsators we have reliable
estimates both for the dynamical part of the gravity (from the radial
velocity) and also for the static part (from the pulsation equation).
Therefore, in principle, one could use a fairly accurate estimate for the
temporal value of the gravity, and determine only $T_{eff}$ and the chemical
abundances. Except for Clementini et al. (1995) and Lambert et al. (1996), we
could not find any other work following this methodology. Consequently, we
have to resort to some method that uses the already published atmospheric
parameters and transforms the temporarily given abundances to the value
corresponding to the static gravity.

Our approach is purely empirical. After experiencing with various data sets,
we found that a linear transformation applied on the published (spectroscopic)
[Fe/H] values yields more stable [Fe/H] values and therefore, supposedly a
better estimate of [Fe/H]222An early reference to the existence of such a
relation can be found in Lambert et al. (1996). See Sect. 4 for further
discussion of their result in the context of this paper.

$\displaystyle{\rm[Fe/H]}_{0}={\rm[Fe/H]}_{sp}+C_{g}(\log g-\log g_{sp})\hskip
5.69054pt,$ (1)

where [Fe/H]0 and [Fe/H]sp, respectively, stand for the transformed and the
direct spectroscopic abundances (with the associated static and spectroscopic
gravities $\log g$ and $\log g_{sp}$). The gravity factor $C_{g}$ is kept
constant at the value to be determined by two different criteria (both seem to
prefer a value near $0.3$ – see Sect. 4).

The static $\log g$ can be easily obtained from linear stellar pulsation
models. Although the single parameter (i.e., period) dependence of the gravity
is very strong, the final formula also depends on the parameter coverage of
the models. The following formula was derived from the models of Kovacs &
Karamiqucham (2021b) covering the RR Lyrae parameter space and combining
solar-scaled models with overall heavy element contents of $Z=0.0008$ and
$0.001$

$\displaystyle\log g=2.48-1.27\log P_{0}\hskip 5.69054pt.$ (2)

This formula fits the model logg values with $\sigma=0.028$, appropriate for
the purpose of correcting the gravity effect in [Fe/H].

In addition to the gravity effect, different metallicity surveys do not use
the very same methodology, including codes, spectral line list, turbulent
velocity, temperature scale, solar metallicity, etc. As a result, there are
author/source-dependent overall zero point (ZP) differences among the
different studies. Furthermore, Eq. 1 is intended only to minimize the gravity
dependence, but this does not guarantee that the resulting [Fe/H] is also
close enough to the static value. To consider both of these effects, we employ
source-by-source differential corrections in the following sense

$\displaystyle[Fe/H](i)=[Fe/H]_{0}(i)+DZP(i)\hskip 5.69054pt,$ (3)

where [Fe/H]${}_{0}(i)$ is the gravity-corrected abundance for some star in
the $i$-th source. Because the methodology followed by Clementini et al.
(1995) and Lambert et al. (1996) seems to be the closest to the type of method
we advocate, the DZP-s are determined in a way which considers the averages of
all available metallicities, but these are additionally shifted (uniformly) by
an amount of $-0.08$ dex that results the smallest ZP corrections for the
metallicities from the above two publications.333Even though these two
publications are based on very close principles, the resulting abundances for
the three common stars differ by $0.1$–$0.2$ dex. This paper uses the average
of the Fe I and Fe II values in the ‘photometric’ section of Table 3 from
Lambert et al. (1996) and Table 12 of Clementini et al. (1995).

To sum it up, for any fixed $C_{g}$, the algorithm of transforming the
published spectroscopic abundances to a uniform scale that observes the start-
by-star $\log g$ and the survey-by-survey ZP dependences, constitutes the
following steps:

  * 1.

Gravity correction: use Eqs. 1,2 for all stars in all sources.

  * 2.

Estimate the ‘ridge’: use each star with multiple sources and compute simple
averages from the logg-corrected [Fe/H]0 values.

  * 3.

ZP corrections: Compute DZP for each source as given by Eq. 3 by considering
the average difference between the ridge and the source values. Add $-0.08$
dex as a global correction to each DZP.

  * 4.

Final [Fe/H]: Loop back to step 1., and derive the final metallicities FEH by
employing the gravity correction on the ZP-corrected input [Fe/H]sp values and
perform simple averaging for objects with multiple measurements.

The above core scheme is run for the scanned $C_{g}$ values in the process to
be aimed at the optimization of $C_{g}$ by using various minimization criteria
for the RMS of the resulting metallicities (see Sect. 4).

##  3 Data sets

In gathering high-quality [Fe/H] data, first we made a search for publications
with simultaneous $T_{eff}$, $\log g$ data based on high dispersion
spectroscopic (HDS) observations. Then, these data were further extended by
big survey data (LAMOST and GALAH). Although LAMOST does not use a HDS
equipment, the method employed is very similar to those of the HDS small-scale
studies. Therefore, for simplicity, we label these data also as HDS. The final
list of HDS data is based on an additional cross-check of the above big survey
databases with the list of Muraveva et al. (2018). We recall that only RRab
variables are included in all these and subsequent data sets.

Table 1: HDS inventory

Source | $N_{obj}$ | $N_{sp}$ | DZP | RMS | $N_{clip}$
---|---|---|---|---|---
ta22 | 21 | 31 | $-0.1053$ | 0.0596 | 0
cr21 | 121 | 184 | $-0.0511$ | 0.0808 | 2
la96 | 15 | 15 | $0.0572$ | 0.0466 | 0
li13 | 22 | 33 | $-0.1654$ | 0.0497 | 0
ch17 | 28 | 70 | $-0.1383$ | 0.0998 | 0
cl95 | 10 | 10 | $-0.0471$ | 0.0966 | 0
fo11 | 11 | 165 | $-0.0601$ | 0.0193 | 0
ne13 | 24 | 24 | $-0.1376$ | 0.1175 | 0
lam5 | 119 | 216 | $-0.0984$ | 0.0798 | 0
lal5 | 81 | 133 | $-0.0395$ | 0.0761 | 1
gala | 22 | 22 | $-0.1629$ | 0.0893 | 0
an18 | 26 | 51 | $-0.0852$ | 0.0882 | 0
pa15 | 18 | 54 | $-0.1378$ | 0.1077 | 0
an21 | 7 | 7 | $-0.1769$ | 0.0838 | 0
fe96 | 9 | 9 | $0.0942$ | 0.1085 | 0
fe97 | 38 | 38 | $-0.0808$ | 0.1495 | 1
so97 | 23 | 47 | $-0.1349$ | 0.1235 | 2
Total: | 269 | 1109 | — | — | 6

Notes: $N_{obj}=$ number of objects; $N_{sp}=$ number of spectra; DZP$=$ zero
point shift for [Fe/H] (including differential and global shifts – see Sect.
2); RMS$=$ standard deviation of the final metallicities for the residuals
between the source items and the averages of the values obtained from all
sources with multiple spectra; $N_{clip}=$ Number of $3\sigma$-clipped
objects.

References: ta22=Takeda (2022); cr21=Crestani et al. (2021); la96=Lambert et
al. (1996); li13=Liu et al. (2013); ch17=Chadid et al. (2017); cl95=Clementini
et al. (1995); fo11=For et al. (2011); ne13=Nemec et al. (2013); lam5=Xiang et
al. (2019); lal5=Luo et al. (2019a, b); gala=Buder et al. (2021);
an18=Andrievsky et al. (2018); pa15=Pancino et al. (2015); an21=Andrievsky et
al. (2021); fe96=Fernley & Barnes (1996); fe97=Fernley & Barnes (1997);
so97=Solano et al. (1997)

Some details of the HDS inventory used are shown in Table 1. We also checked
two more sources: Sprague et al. (2022), based on the APOGEE survey and
Gilligan et al. (2021), from the observations made by SALT/HRS. Unfortunately,
both of these sets proved to be too noisy with respect of the sources listed
in Table 1.

As a consistency check, we also examined the earlier low dispersion
spectroscopic (LDS) data by Layden (1994) and Suntzeff et al. (1994)
(hereafter LA94 and SU94, respectively). We recall that the LDS metallicities
are based on the calibration of spectral indices (or equivalent widths) that
are conveniently measured also on low dispersion spectra. These data are then
calibrated on proper sets of [Fe/H] derived from HDS spectra (with the HDS
methodology). Once that LDS parameters have been calibrated, they can be
quickly applied to determine the metallicities of new objects, often not
accessible by HDS instruments. Somewhat surprisingly, we found that these
early sets follow quite well the HDS ridge (star-by-star average [Fe/H], see
Fig. 1). Both sets show systematic differences with respect to the HDS values.
We approximate these trends by straight lines. The regression coefficients are
given in Table 2. The improved quality of the merged (HDS & LDS) data will be
demonstrated in Sect. 5.

Concerning the calibration of the gravity- and source ZP-corrected [Fe/H]
relative to the Sun, we note that an overwhelming majority of the sources use
$\log\epsilon(Fe)=7.50$. Therefore, [Fe/H] presented in this work is
considered to be tied to this solar value.

Table 2: LDS to HDS transformations

Source | $c_{1}$ | $c_{2}$ | RMS | $N$ | $N_{clip}$
---|---|---|---|---|---
LA94 | 0.0216 | 0.9433 | 0.1740 | 184 | 8
SU94 | 0.2514 | 1.1147 | 0.2359 | 64 | 0

Notes: [Fe/H]${}_{HDS}\sim c_{1}+c_{2}$[Fe/H]LDS; the fit was made to FEH
(gravity- and source ZP-corrected HDS [Fe/H]); $N_{clip}=$ number of
$3\sigma$-clipped objects. See accompanying Fig. 1.

Figure 1: HDS ridge metallicities (FEH) vs individual HDS (gray dots) and LDS
(black dots) metallicities. The corresponding regression lines are shown by
gray and black lines. The standard deviations of the LDS regressions are shown
in the upper left corners of the corresponding LDS sources: LA94 for Layden
(1994) and SU94 for Suntzeff et al. (1994).

##  4 The gravity factor

Before dwelling more deeply into the gravity correction, it is worth examining
the distribution of the published gravities of the various surveys and
targeted studies. Figure 2 shows the distribution of the gravity differences
$\Delta\log g=\log g-\log g_{sp}$ for the various sources. Large deviations
can be seen in both directions, suggesting improper pulsation phasing of the
observations, or, more likely, large errors in the spectroscopic gravities.
The surprisingly flat pattern of $\log g$ for the set of fo11 (For et al.,
2011) is due to their specific sample of stars, with periods concentrated in a
narrow range. The striking linear dependence for fe96, fe97 and so96
(respectively, Fernley & Barnes, 1996, 1997; Solano et al., 1997) is due to
the fixed $\log g_{sp}$ values in their analysis. The $\Delta\log g$ ranges
are particular small and quite close to the values predicted by Eq. 2 for la96
and cl95 (respectively, Lambert et al., 1996; Clementini et al., 1995),
indicating proper phasing of the observations in these studies.

Figure 2: Gravity differences (static value, $\log g$, from Eq. 2 minus
published spectroscopic value) for the individual objects of the different
sources. For each source, a uniform vertical shift was applied to separate it
from the data of other sources. See Table 1 for the source acronyms.

After a detailed examination of the star-by-star dependence of the published
[Fe/H] values on gravity difference $\log g-\log g_{sp}$, we found that the
gravity factor $C_{g}$ (Eq. 1) has a visible target dependence. For instance,
Z Mic has remarkably constant [Fe/H] implying $C_{g}\sim 0.0$, whereas V1645
Sgr is best fitted by a steep $C_{g}$ of $\sim 0.6$. However, using individual
$C_{g}$ would lead to a high level of overfitting, due to the modest number of
spectra for most of the stars.444Nevertheless, for further support of our
adopted value of $C_{g}$, it is useful to examine the distribution of the
star-by-star fitted $C_{g}$ values and check the degree of spread of these
values. See Appendix C for this test. Therefore, we decided to search for the
best global value of $C_{g}$, by minimizing the scatter of the $\log
g$-transformed and source ZP-adjusted abundances around the average values for
all stars with multiple spectra ($197$ stars and $1037$ spectra altogether).
Black dots in Fig. 3 show the run of RMS (individual transformed metallicities
{feh} minus their average FEH) for the full scan range of $C_{g}$. A
preference for $C_{g}\sim 0.2$ is clearly visible.

Yet another way to optimize $C_{g}$, is to scan the dependence of the RMS of
the $(P,\varphi_{31})\rightarrow$ FEH fit (Jurcsik & Kovacs, 1996), this time
using the full HDS set, including single spectra data (263 objects with
available $V$ light curves – see Sect. 5). Gray dots in Fig. 3 indicate a
preference of high significance for a global value of $C_{g}\sim 0.3$.

Figure 3: Dependence of the fit RMS values on the gravity correction factor
$C_{g}$ (see Eq. 1). The plot shows the scans resulting from the direct
estimates of the scatter of the individual transformed abundances {feh} around
the average values {FEH} and the variation of the fit RMS of the Fourier-based
estimates. The shaded box indicates the optimum regime for $C_{g}$.

To further constrain $C_{g}$, we also attempted to test the tightness of the
[Fe/H]$-M_{V}$ (e.g., Castellani et al., 1991) and the near infrared period-
luminosity-metallicity (PLZ, i.e., Bono et al., 2003) relations.
Unfortunately, both of these relations have relatively weak metal dependences.
In addition, this dependence is further masked by noise, internal physical
scatter (for the [Fe/H]$-M_{V}$ relation) and the correlation between the
period and metallicity (for the PLZ test). Therefore, we rely only on the
direct tests shown in Fig. 3. These tests suggest values of $C_{g}\sim 0.25$,
which we will use throughout the paper.

For a very simple demonstration of the differential $\log g$-dependence of the
published spectroscopic abundances, in Fig. 4 we plot these [Fe/H] values as a
function of $\Delta\log g=\log g-\log g_{sp}$ (with $\log g$ as given by Eq.
2). In the left panel, all [Fe/H] were taken straight from the respective
publications and plotted against $\Delta\log g$. In the right panel we
employed source ZP corrections as described in Sect. 2 and presented in Table
1. For each star, the [Fe/H]sp values were fitted by a straight line of Eq. 1
with $C_{g}=0.25$ and [Fe/H]0 adjusted by a least-squares fit of equal
weights.

As noted earlier, although in most cases the adopted slope is in good
agreement with the observations, there are cases with considerable differences
(steeper, like V1645 Sgr and shallower, such as AN Ser). Source ZP corrections
further improve the fit as can be seen by the insets at each object showing
the RMS of the residuals around the straight lines.

Figure 4: Examples of the star-by-star correlation between the gravity
difference and the published abundances (left panel). The same correlation is
shown in the right panel after source ZP corrections. For an easier
visualization, vertical shifts were applied together with a factor of two
decrease of the [Fe/H] ranges (including the slope of the fitted straight line
with an original value of $-0.25$). Numbers at the individual objects show the
RMS values around the straight lines). See text for further details. Figure 5:
Left: Correlation between the published HDS metallicities and the difference
between the static and spectroscopic gravities for the $197$ stars with
multiple spectra. Vertical shifts were applied according to Eq. 1 to avoid
scatter due to differences in the stellar metallicities. Right: As in the left
panel, but only for two well-fit examples. Triangles are for the RRab sample
from Table 3 of Lambert et al. (1996) by using the ‘photometric’ and
‘spectroscopic’ results (see text for further details).

We can also visualize the $\Delta\log g\rightarrow$[Fe/H] correlation by
collapsing the data in the vertical axis (i.e., not applying star-by-star
vertical shifts). The left panel of Figure 5 shows the result of this type of
data handling by using the full sample of $197$ stars with multiple
measurements. In the right panel, in addition to showing two well-fit cases we
also exhibit the specific data from Lambert et al. (1996). To the best of our
knowledge their work is the only one (in the context of RR Lyrae stars)
comparing metallicities from the traditional spectroscopic method and those
obtained by using known values of the temporal gravity and theoretical
spectral models.

Using Table 3 of Lambert et al. (1996), first we calculated the average of the
‘photometric’ metallicities derived from the FeI and FeII lines.555Although
the correlation with $\Delta\log g$ is tighter for FeII (Lambert et al.,
1996), we used the average for consistency with the final data set of this
paper. These metallicities were obtained by employing the $\log g$ values
mostly from various Baade-Wesselink analyses, and, in fewer cases, from
narrow-band photometry. Then, we used the ‘spectroscopic’
metallicities666These are derived from direct theoretical spectrum fits, based
on equating the abundances obtained from spectral lines of various ionization
levels. The gravity is computed as part of the fitting process. to calculate
the difference between the these and the ‘photometric’ metallicities. The
corresponding gravity differences (‘spectroscopic’ minus ‘photometric’) are
also listed in the same table of Lambert et al. (1996). Except for a vertical
adjustment of $-0.09$ dex, we plotted straight these published values in Fig.
5. The $15$ RRab stars follow remarkably well the trend observed by the other
two individual objects with multiple measurements.

Although it is not easy to find a single source for the correlation between
$\Delta\log g$ and [Fe/H], non-LTE effects are likely play an important role
(Lambert et al., 1996). The work by Luck & Lambert (1985) on intermediate-mass
supergiants (including Cepheids) was the first to point out the systematic
differences between the gravities obtained from the spectroscopic analyses and
pulsation models. Being less sensitive to non-LTE effects, Kovtyukh &
Andrievsky (1999) suggest using Fe II lines for abundance and gravity
estimates in the case of Cepheids. Lambert et al. (1996) lend on a similar
conclusion for RR Lyrae stars.

To visualize the improvement of the quality of the derived metallicities at
various stages of the process, in Fig. 6 we show the star-by-star standard
deviations for the above set of 197 stars. The ZP correction yield an RMS
improvement which is significant at the $2\sigma$ level. This is quite close
to the additional $2.3$ sigma significance of the gravity correction on the
ZP-corrected values. All these indicate essentially the same level of
importance of both types of correction.

Figure 6: Ordered standard deviations of the multiple [Fe/H] values under
various data handling as given by the code list in the upper left part of the
figure. First number in the code symbol corresponds to the logg correction
term ($C_{g}$ in Eq. 1), whereas the second number indicates if source-by-
source zero point correction was (1) or was not (0) employed. The inserted
panels show the scatter of the star-by-star individual metallicity values
{feh} around the corresponding average metallicities FEH. The equality lines
are indicated by light blue. The standard deviations are shown by the internal
labels.

##  5 The $(P,\varphi_{31})\rightarrow$[Fe/H] fit

Here we test further the gravity- and source zero point-corrected
metallicities FEH (Sects. 4 and 3). This test also casts more light on the
potentials and limitations of the metallicity determination based on light
curve analyses (Kovacs & Zsoldos, 1995; Jurcsik & Kovacs, 1996).
Unfortunately, this relation still remains to be purely empirical, without any
theoretical foundation (except perhaps for the work of Feuchtinger, 1999), in
spite of the practical applicability and current recalibrations on various
data sets (e.g., Dékány & Grebel, 2022; Li et al., 2023).777At the same time,
it is worthwhile to note that the large datasets used in this paper further
strengthen the strong preference for the $(P,\varphi_{31})$ dependence against
any other two-parameter combinations, including $\varphi_{41}$ and $A_{tot}$
(depending on the data sets, at the $1-4$ and $8-12$ $\sigma$ levels,
respectively).

We use the light curves from the ASAS and ASAS-SN surveys (Pojmanski, 1997;
Shappee et al., 2014; Christy et al., 2023) and the Fourier decompositions
presented by Jurcsik & Kovacs (1996). Because our approach here is to utilize
as many spectroscopic [Fe/H] as possible, we include all RRab stars,
irrespectively if they are monomode or Blazhko. If the target is of this
latter type, it is frequency analyzed, and the mid curve (i.e., that part of
the light curve that comes from the monomode pulsation) is used to estimate
the Fourier phase. Additional details of the light curve analysis can be found
in Appendix B.

The metallicity data sets to be dealt with are listed in Table 3. These sets
were selected to investigate: (i) effect of merging LDS and HDS data to aim
for more accurate test base (set A, containing stars with average
metallicities derived from HDS and LDS sources); (ii) more realistic
situations, when the data quality strongly changes within the sample (set B,
with mixed items containing only either HDS or LDS, or type A metallicities).
The RMS values indeed show that the merged data yield better result. Assuming
Gaussian distributions for the residuals (an assumption that has only partial
validity – see later), we get that the RMS differences with respect to set A
for AHDS, ALDS and B are significant at the $1.7$, $2.5$ and $1.4$ sigma
levels, respectively. For set AHDS, employing only HDS source ZP corrections
(i.e., no gravity correction), we get a fit with an RMS of $0.256$. With
respect to set A, this increase is significant at the level of $5.7$ sigma.

Table 3: Datasets for the $(P,\varphi_{31})\rightarrow$[Fe/H] test

Name | NFEH | Content | RMSfit | Nclip
---|---|---|---|---
A | 187 | HDS with LDS | 0.163 |  03
AHDS | 187 | HDS only in A | 0.186 |  02
ALDS | 187 | LDS only in A | 0.199 |  01
B | 390 | A$+$HDS only$+$LDS only | 0.179 | 11

Figure 7: Metallicity predictions FEH31 from the period and Fourier parameter
$\varphi_{31}$. Datasets given in Table 3 are used including both $\log g$\-
and ZP-corrected HDS data and LDS data as discussed in Appendix A. These
metallicities are shown on the horizontal axis. Straight lines indicate the
equality values for [Fe/H]. The plot for set B has been shifted downward for
better visibility.

The corresponding regressions for sets A and B are shown in Fig. 7. We recall
that the samples shown include all stars (i.e., even those indicated as
‘clipped’ objects in Table 3. Systematically high-fit objects at the low
metallicity end are clearly visible. Assuming that the Fourier method is
sensitive to the global metallicity [M/H] rather than to [Fe/H], then, this
effect may come from the stronger $\alpha$ element enhancements at lower
metallicities. On the other hand, we see also systematically low-fit values
for quite a large number of stars in the mid- to high-metallicity part. A
brief test of the effect of possible low (or perhaps negative) $\alpha$
enhancement for these stars by using the estimates of Crestani et al. (2021)
did not suggest any statistically significant improvement. A more detailed
examination of these stars vaguely implicated that the underestimation by the
Fourier method may be partially attributed to the slight excess of Blazhko
stars (however, we do not have an explanation why these stars may cause such
an effect).

In addition to the visual inspection of the residuals, we can also examine
their distribution. It turns out (see Appendix D) that none of the data sets
follow Gaussian distribution in the outskirt of their ordered residuals. About
$20$% of the stars in each sample belong to a non-Gaussian subset. Although
several outliers can be explained by light curve or metallicity
errors/peculiarities, both in the non-Gaussian set and also among the simple
outliers, there are stars with excellent metallicities and accurate light
curves. Stars such as X Ari, AL CMi or V0341 Aql remain among the puzzling
details of the Fourier method for [Fe/H] determination.

From purely practical point of view, a large fraction of the observed [Fe/H]
values are fitted remarkably well with $(P,\varphi_{31})$. The errors are less
than $0.2$ dex for over $70$% and less than $0.1$ dex for about $50$% of the
stars in our samples.

Finally, for $V$ light curves (using the monomode components for Blazhko
stars) the updated formula derived from set A of Table 3 reads as follows888By
using set B we get $FEH_{31}=-4.845-5.385\,P+1.274\,\varphi_{31}$, yielding
estimates that deviate from the values predicted by Eq. 4 with a standard
deviation of $0.017$ dex.

$\displaystyle FEH_{31}=-5.088-5.268\,P+1.311\,\varphi_{31}\hskip 5.69054pt.$
(4)

##  6 Conclusions

By using the best available iron abundances based on traditional spectral
analyses of fundamental mode RR Lyrae stars, we arrived to the following
conclusions.

  * $-$

Current spectroscopic abundances suffer from considerable scatter both
internally (within a given survey) and externally (between the different
surveys).

  * $-$

The internal scatter can be mitigated on a star-by-star basis by employing a
linear correction including the difference between the static and the temporal
(spectroscopic) gravities.

  * $-$

The external (zero point) differences can be eliminated by uniform survey-by-
survey shifts based on the common stars in the different surveys.

  * $-$

Both corrections yield roughly the same degree of improvement in the overall
quality of the final abundances, resulting in a RMS of $0.12$ dex around the
averages for stars with multiple measurements. This is an improvement above
$4$ sigma with respect to the RMS of $0.17$ dex of the published (uncorrected)
values.

The relatively large differences among the spectroscopic abundances have led
us to examine the possible utilization of the metallicity estimates based on
spectral index methods (Layden, 1994; Suntzeff et al., 1994). Interestingly,
we found these metallicities quite comparable with the overall higher accuracy
direct spectroscopic metallicities. The merged set of $187$ stars (set A in
Table 3) represents the most accurate metallicities used in this paper.

The higher quality [Fe/H] reflects also on the tightness of the relations
involving [Fe/H]. Although the effect is rather mild on the [Fe/H]$\rightarrow
M_{V}$ and near infrared $(logP,{\rm[Fe/H]})\rightarrow M_{Ks}$ relations, the
improvement on the $(P,\varphi_{31})\rightarrow$[Fe/H] fit is far more
significant. In this update we extended the applicability of the Fourier
method to Blazhko stars by using the average light curves, as derived from the
Fourier fits including also the Blazhko modulation. The precision of the
[Fe/H] estimates are, respectively, within $0.2$ and $0.1$ dex for some $70$%
and $50$% of the samples investigated. The Fourier-based [Fe/H] yield a little
tighter correlations for the above luminosity relations. At the same time, we
found puzzling differences between the Fourier estimates and the observed
values in several stars. We cannot offer any explanation for this phenomenon
at this moment, except perhaps for the vague idea that the Fourier estimate
refers more likely to the overall metal abundance rather than [Fe/H] and that
for some reasons some of the stars are $\alpha$ element deficient (rather than
enhanced).

Based on the improvement presented in this paper, we strongly argue for the
reexamination of methodology of element abundance determination in RR Lyrae
stars, and in pulsating stars, in general. The method used by Clementini et
al. (1995) and Lambert et al. (1996) are important examples for the path we
think would be very fruitful to pursue.

###### Acknowledgements.

We appreciate the discussion with Giuliana Fiorentio on the intricacy of
$\alpha$-enhancement. It is a pleasure to thank the useful comments by
Jayasinghe A. Tharindu on the compatibility of the ASAS and ASAS-SN light
curves. We thank the referee for drawing our attention on some earlier works
relevant for this paper. This research has made use of the VizieR catalogue
access tool, CDS, Strasbourg, France (DOI: 10.26093/cds/vizier). This work has
made use of data from the European Space Agency (ESA) mission Gaia
(https://www.cosmos.esa.int/gaia), processed by the Gaia Data Processing and
Analysis Consortium (DPAC,
https://www.cosmos.esa.int/web/gaia/dpac/consortium). Funding for the DPAC has
been provided by national institutions, in particular the institutions
participating in the Gaia Multilateral Agreement. This research has made use
of the NASA/IPAC Infrared Science Archive, which is funded by the National
Aeronautics and Space Administration and operated by the California Institute
of Technology. This research has made use of the International Variable Star
Index (VSX) database, operated at AAVSO, Cambridge, Massachusetts, USA.
Guoshoujing Telescope (the Large Sky Area Multi-Object Fiber Spectroscopic
Telescope LAMOST) is a National Major Scientific Project built by the Chinese
Academy of Sciences. Funding for the project has been provided by the National
Development and Reform Commission. LAMOST is operated and managed by the
National Astronomical Observatories, Chinese Academy of Sciences. This work
made use of the Third Data Release of the GALAH Survey (Buder et al., 2021).
Supports from the National Research, Development and Innovation Office (grants
K 129249 and NN 129075) are acknowledged.

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##  Appendix A Metallicities

Spectroscopic [Fe/H] abundances based on HDS methodology were collected in a
multistep process, whereby the additional data sets were iteratively examined
for quality control. This process has led to re-instating sources that were
priorly excluded and vice verse. Unfortunately, we had to exclude two sources
(Gilligan et al., 2021; Sprague et al., 2022) due to excessive scatter around
the [Fe/H] ridge established by the other sources. Examples on a well-fit
subset and one of the excluded sets are shown in Fig. 8.

Figure 8: Source-dependent individual vs ridge metallicities (FEH, averages
computed from all sources). Black line shows the ridge, blue line shows the
fit to the particular source averages (assuming only a constant shift relative
to the ridge line for a given source). Black dots are for the average
metallicities, pale, larger circles are for the individual metallicities. The
left and right panels show, respectively, the well-fit case of Luo et al.
(2019a, b) (LAMOST DR5, DR7) and the poor-fit case of Gilligan et al. (2021).
The latter is not included in our final HDS sample. Table 1 is referred for
the list of the HDS sources.

Finally, the set of 17 sources was established, with $1279$ individual
observations, often multiple visits to the same objects. Among the targets
there were many RRc stars, several RRd, BL Her and binary stars. These,
together with faint ($V\gtrsim 15$ mag) objects (of any type) were excluded
(altogether $104$ objects) from further analysis. The final inventory of $269$
RRab stars is shown in Table 1.

Concerning the data handling, the following points worth mentioning. For the
“big survey” data (LAMOST and GALAH) an upper error limit of $0.4$ dex was
employed. After a considerable amount of testing, we decided to use simple
averaging instead of robust averaging for the computation of the final
metallicities for objects with multiple sources. This choice was justified
because of the overall moderate number of multiple measurements for the
individual objects. In merging the HDS and LDS data we considered the
individual measurements in both sets to be equal, therefore, the final [Fe/H]
was computed as a weighted average, i.e.,
[Fe/H]$=w$[Fe/H]${}_{HDS}+(1-w)$[Fe/H]LDS, where
$w=N_{HDS}/(N_{HDS}+N_{LDS})$, and the $N$-s stand for the number of spectra.
Except for the following two stars, all objects were treated equally. We did
not use the HDS values of V0455 Oph, because there were differences up to
$0.9$ dex among the four published values. For WW Vir, the value of Xiang et
al. (2019) was an extreme outlier with respect of the values of Luo et al.
(2019a, b) and Layden (1994), so, Xiang’s et al. value was omitted.

##  Appendix B Light curves

We opted to employ the long-term V-band time series photometry supplied by the
ASAS and ASAS-SN surveys. These projects yield unique data sets fitting
perfectly to our goals to derive: (i) reliable low-order phases for testing
the Fourier-based [Fe/H] estimation; (ii) homogeneous sets of V magnitude
averages for testing the $M_{V}$ dependence on [Fe/H]; (iii) proper
decomposition of the light curves into the monomode and Blazhko components for
goal (i) and for the assessment of the role of Blazhko phenomenon in the
applications.

For variables without an apparent Blazhko modulation, we employed standard
least squares Fourier fit up to order $15$ (depending on the quality of the
light curve). Outliers were iteratively clipped. For the Blazhko stars, first
we performed a scan of the Blazhko period by using the following simple
Fourier model, considering only the first order modulation side lobes

$\displaystyle V(t)$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{m}A_{i}^{-}\sin(\omega_{i}^{-}t+\varphi_{i}^{-})+A^{+}_{i}\sin(\omega_{i}^{+}t+\varphi_{i}^{+})$
(5) $\displaystyle+$
$\displaystyle\sum_{j=1}^{k}A_{j}\sin(\omega_{j}t+\varphi_{j})\hskip
5.69054pt,$

where $m$ is the highest harmonics with side lobe frequencies $\omega_{i}^{-}$
and $\omega_{i}^{+}$. The Blazhko frequency is
$\omega_{B}=\omega_{i}-\omega_{i}^{-}=\omega_{i}^{+}-\omega_{i}$, with
$\omega_{i}$ being the $i^{th}$ harmonics of the fundamental frequency. We
used $m=7$ in nearly all cases and constrained the monomode component to order
$k\geq m$. Although the frequency spectra of Blazhko stars are, in general,
more complicated than the simple model above, in a large majority of cases it
yielded a good description of the data.

For the Fourier phases and average magnitudes, in addition to ASAS and ASAS-
SN, we considered also the data collected by Jurcsik & Kovacs (1996) based on
classical individual stellar studies. The final set of $\varphi_{31}$ and the
magnitude averages (V), result from the zero point shifted equal-weighted
averages from all possible sources (by equal weighting all three sources). The
zero point shifts – Jurcsik & Kovacs (1996) minus source – for $\varphi_{31}$
are $-0.015$ and $+0.015$ for ASAS and ASAS-SN, respectively. The shifts for
(V) – in the same order and same sense – are as follows: $+0.016$ and $+0.039$
mag, with high respective standard deviations of $0.013$ and $0.037$.

There are two stars that have not been used in any light curve-related tests.
The bright Blazhko stars RR Lyr and XZ Cyg are only in the ASAS-SN database,
but they are saturated, with XZ Cyg having too few data points. TZ Aur is
monoperiodic, but older data were fractional, resulting questionable Fourier
phases. Therefore, the star was not used from Jurcsik & Kovacs (1996).

##  Appendix C Distribution of the gravity coefficient

As noted in Sect. 4, individual cases might yield quite different values for
the gravity factor $C_{g}$. To get an overall view on the statistics of the
$C_{g}$ values, for each star, we fit the published [Fe/H] values by an equal-
weighted least squares method using the following formula

$\displaystyle{\rm[Fe/H]}_{sp}=a_{0}+a_{1}\Delta\log g\hskip 5.69054pt,$ (6)

where $\Delta\log g=\log g-\log g_{sp}$, with $\log g$ denoting the estimated
static (Eq. 2) and $\log g_{sp}$ the published spectroscopic gravities. In
this equation coefficient $a_{1}$ corresponds to the object-dependent $C_{g}$
(with a negative sign, due to the re-arrengement of Eq. 1). The distribution
of {$a_{1}$} is shown in Fig. 9. We see that the distribution is asymmetric
around the zero slope and peaks close to our adopted value of $a_{1}=-0.25$.
The plot shown was derived from the dataset obtained by selecting objects with
more than two spectroscopic measurements. The dependence of the averages of
{$a_{1}$} on the allowed minimum number of spectra is given in Table 4. We
used $3\sigma$ iterative clipping in each case to derive the averages and the
corresponding RMS values. We note that the finally adopted value of $C_{g}$
weights also on the more negative value preferred by the Fourier method (see
Fig. 3).

Figure 9: Probability distribution function of the gravity coefficient $a_{1}$
in the linear regression of $\Delta\log g$ to the published abundances (see
Eq. 6). For reference, dotted and thick vertical lines, respectively, show the
$a1=0.0$ and our final adopted value $a1=-0.25$.

Table 4: Average gravity coefficient as a function of minimum number of
spectra

$Nsp_{min}$ | $\langle a_{1}\rangle$ | $RMS(a_{1})$ | $N$ | $N_{clip}$
---|---|---|---|---
2 | -0.217 | 0.194 | 194 | 20
3 | -0.222 | 0.173 | 127 | 7
4 | -0.210 | 0.148 | 108 | 8
5 | -0.208 | 0.147 | 72 | 3

Notes: $Nsp_{min}$: minimum number of spectra per object; $\langle
a_{1}\rangle$: average of {$a_{1}$}; $RMS(a_{1})$: RMS of {$a_{1}$}; $N$:
Number of stars; $N_{clip}$: Number of stars clipped.

##  Appendix D Associated results

Here we briefly summarize some of the consequences of the metallicity scale
introduced in this paper. We focus on the residual distribution of the
metallicity estimate based on the Fourier parameters, the classical
metallicity–absolute V magnitude and the near infrared PLZ relations.

A general property of all metallicity sets studied in this paper is the non-
Gaussian distribution of the subsets of stars lying in the gray zone of few
sigma deviations with respect to the best linear fit of [Fe/H]$\sim
a_{0}+a_{1}P+a_{2}\varphi_{31}$. To demonstrate the size of the effect, we use
set A of Table 3 (see also Fig. 7 and Eq. 4 for the regression). The ordered
residuals are shown in Fig. 10. For some $80$% of the sample, the distribution
is within the $3\sigma$ range of the same size of sets following Gaussian
distribution with a standard deviation of $\sigma_{G}=0.140$.

As mentioned in Sect. 5, currently we have no explanation for this phenomenon.
Once the outliers (both the non-fitting and non-Gaussian members – 14 stars)
are omitted, the resulting sample of 173 stars yields a fit RMS of $0.135$ dex
and a core Gaussian distribution with $\sigma_{G}=0.127$ dex. It is important
to recall, that, while some stars are suspect of light curve anomalies (e.g.,
IU Car [period and light curve changes], SS CVn [Blazhko with peculiar average
light curve]) several stars, qualified as outliers (e.g., AL CMi, X Ari) have
very accurate observed abundances, and no apparent light curve anomalies over
several/many decades.

Figure 10: Testing the $(P,\varphi_{31})\rightarrow{\rm FEH}$ fit residuals.
Inset: Ordered distribution (light blue) of the residuals ($\Delta{\rm
FEH}=|{\rm FEH_{obs}-FEH_{fit}}|$) for set A of Table 3. Black line shows the
distribution of the corresponding Gaussian, fitting the core of the observed
residuals. Main panel: Difference between the observed and the predicted
Gaussian residuals. The $3\sigma$ ranges for the Gaussian residuals are shown
by the vertical error bars. Figure 11: Iron abundance vs V absolute magnitude,
using a subset of set B (see Table 3 and text for the selection of this
subset). Lines are the resulting linear regressions. For better visibility,
the set using Fourier-based abundances (FEH31) has been shifted downward by
$2$ mag with respect of the set using the mixture of gravity/source ZP-
corrected HDS and LDS metallicities.

The effect of the type of metallicity used in studying the metallicity-
absolute magnitude relation is shown in Fig. 11. We used the Gaia DR3
parallaxes (Lindegren et al., 2021) and reddenings by Schlafly & Finkbeiner
(2011). Set B of Table 3 was selected as a test base, but we employed cutoffs
for $E(B-V)$ and for the relative parallax errors of $0.2$ and $0.06$,
respectively. This had led to a sample of $264$ stars. To calculate the
Fourier FEH, we used Eq. 4. This yielded a residual standard deviation of
$0.163$ mag (vs $0.184$ from the fit using direct FEH values). The difference
is marginally significant (at the $2\sigma$ level). By using other samples, we
lend on similar conclusions, often with a lower significance. As expected, the
linear regressions are also very similar, yielding
$M_{V}=1.059+0.341\times{\rm FEH}$ for the direct and
$M_{V}=1.095+0.372\times{\rm FEH_{31}}$ for the Fourier-based FEH31. We also
examined the effect of gravity and source ZP corrections, and found similar
low-significance preference for the corrected metallicities with respect to
the uncorrected ones.

Figure 12: Observed 2MASS absolute magnitude (Ks) vs predicted magnitudes from
various PLZ relations. The lines show the identity values, the labels indicate
the formula type used in deriving the various fits. The inset shows the
correlation between the two quantities fitted to the Ks magnitudes. See text
for sample selection.

Finally, in Fig. 12 we show the PL and PLZ relations in the near infrared. The
goal of this test is to check: (a) the dependence on the metallicity used; (b)
the significance of the metallicity dependence. We use the sample of $\sim
100$ stars with approximate average Ks magnitudes transformed from the unWISE
fluxes (Schlafly et al., 2019) as given by Kovacs & Karamiqucham (2021a).
After cross-matching with set B of Table 3 and applying $E(B-V)$ and relative
parallax error cuts of $0.5$ and $0.03$, respectively, we ended up with a
sample of $55$ stars. As shown in Fig. 12, we get increasingly better
regressions from the single-parameter ($\log P$) fit to those utilizing the
metallicities FEH and FEH31. The latter (best) fit is significantly better
than the single-parameter fit nearly at the level of $3\sigma$. On the other
hand, we found basically no difference between using the original (published,
not corrected) [Fe/H] and the gravity/source-corrected FEH. As shown in the
inset of Fig. 12, this is most probably attributed to the significant
correlation between $\log P$ and [Fe/H] (the period dependence ‘takes over’ if
the available [Fe/H] becomes more noisy, yielding a fit of similar quality).
Indeed, we obtained the following regressions:

$M_{Ks}=-1.4316-3.6331\times\log P$,

$M_{Ks}=-0.9526-2.5676\times\log P+0.1554\times{\rm FEH}$,

$M_{Ks}=-0.8874-2.4436\times\log P+0.1899\times{\rm FEH_{31}}$,

with $\sigma_{fit}=0.090$, $0.067$ and $0.060$, respectively.