Coupling conditions for the $3\times 3$ Euler system

1.   Introduction

Topological indices have become an important research topic associated with the study of their mathematical and computational properties and, fundamentally, for their multiple applications to various areas of knowledge (see, e.g., ^{[2,3,4,5,6,7,8,9]}). Within the study of mathematical properties, we will contribute to the study of the optimization problems involved with topological indices (see, e.g., ^{[10,11,12,13,14,15,16,17,18]}).

In ^[19,20] several degree-based topological indices, called adriatic indices, were presented; one of them is the inverse sum indeg index $IS\!I$. It is important to note that this index was selected as one of the most predictive, in particular associated with the total surface area of the isomers of octane.

Let $G$ be a graph and $E(G)$ the set of all edges in $G$, denote by $uv$ the edge of the graph $G$ with vertices $u, v$ and $d_z$ is the degree of the vertex $z$. the $ISI$ index is defined by

Nowadays, this index has become one of the most studied from the mathematical point of view (see, e.g., ^{[21,22,23,24,25,26,27]}). We study, here, the mathematical properties of the variable inverse sum deg index defined, for each $a \in \mathbb{R}$, as

Note that $IS\!D_{-1}$ is the inverse sum indeg index $ISI$.

This research is motivated, in general, by the theoretical-mathematical importance of the topological indices and by their applicability in different areas of knowledge (see ^[28,29,30]). Additionally, in particular, by the work developed by Vukičević entitled "Bond Additive Modeling 5. Mathematical Properties of the Variable Sum Exdeg Index" (see ^[1]), where several open problems on the topological index $IS\!D_a$ were proposed. The novelty of this work is given in two main directions. The first one is associated with the solution of some of the problems posed in ^[1]. The second one is associated with the development of new optimization techniques and procedures related to the monotony and differentiation of symmetric functions, which allowed us to solve extremal problems and to present bounds for $IS\!D_a$. Although, these techniques can be extended or applied in a natural way to obtain new relations and properties of other topological indices, it should be noted that their applicability requires the monotony of the function that determines the index to be studied.

In Section 2, we find optimal bounds and solve extremal problems associated with the topological index $IS\!D_a$, with $a < 0$, for several families of graphs. In Proposition 4, we solve the extremal problems for connected graphs with a given number of vertices. Theorem 6 and Remark 1 solve these problems for graphs with a given number of vertices and minimum degree; similarly, Theorems 8 and 9 present solutions to extremal problems in connected graphs with a given number of vertices and maximum degree. In this direction, in Theorem 5, Proposition 7 and Theorem 10, we present optimal bounds for the studied index.

In Section 3 of this research, a QSPR study related to the $IS\!D_a$ index in polyaromatic hydrocarbons is performed using experimental data. First, we determine the value of $a$ that maximizes the Pearson's correlation coefficient between this index, and each of the studied physico-chemical properties. Finally, models for these properties are constructed using the simple linear regression method. A discussion of the results obtained is presented in Section 4, and some open problems for future research on this topic are raised.

In this research, $G = (V (G), E (G))$ denotes an undirected finite simple graph without isolated vertices. By $n$, $m$, $\Delta$ and $\delta$, we denote the cardinality of the set of vertices of $G$, the cardinality of the set of edges of $G$, its maximum degree and its minimum degree, respectively. Thus, we have $1 \le \delta \le \Delta < n$. We denote by $N(u)$ the set of neighbors of the vertex $u \in V(G)$.

2.   Extremal problems

Suppose $\delta < \Delta$, we say that a graph $G$ is $(\delta, \Delta)$-quasi-regular if it contains a vertex $w$, such that $\delta = d_w$ and $\Delta = d_z$ for every $z \in V(G) \setminus \{w\}$; $G$ is $(\delta, \Delta)$-pseudo-regular if it contains a vertex $w$, such that $\Delta = d_w$ and $\delta = d_z$ for every $z \in V(G) \setminus \{w\}$.

In ^[31] appears the following result.

Lemma 1. Let $k$ be an integer, such that $2 \le k < n$.

$(1)$ If $n k$ is even, then there exists a $k$-regular graph that is connected and has $n$ vertices.

$(2)$ If $n k$ is odd, then there exist a $(k, k-1)$-quasi-regular and a connected $(k+1, k)$-pseudo-regular graphs, which are connected and have $n$ vertices.

The following result is basic to the development of this work.

Lemma 2. For each $a < 0$, the function $f: \mathbb{R}^+ \times \mathbb{R}^+ \rightarrow \mathbb{R}^+$ given by

is strictly increasing in each variable.

Proof. Since $a < 0$, we have

Then, $f$ is a strictly increasing function in $x$, and since $f$ is symmetric, it is also strictly decreasing in $y$.

Using Lemma 2, we obtain the following result.

Proposition 3. If $G$ is a graph, $u, v \in V(G)$ with $uv \notin E(G)$, and $a < 0$, then $IS\!D_a(G \cup \{uv\}) > IS\!D_a(G)$.

Given an integer number $n\ge 2$, let $\mathcal{G}(n)$ (respectively, $\mathcal{G}_c(n)$) be the set of graphs (respectively, connected graphs) with $n$ vertices.

Next, given integer numbers $1 \le \delta \le \Delta < n$, we are going to define the following classes of graphs: let $\mathcal{H}(n, \delta)$ (respectively, $\mathcal{H}_c(n, \delta)$) be the graphs (respectively, connected graphs) with $n$ vertices and minimum degree $\delta$, and let $\mathcal{I}(n, \Delta)$ (respectively, $\mathcal{I}_c(n, \Delta)$) be the graphs (respectively, connected graphs) with maximum degree $\Delta$ and $n$ vertices.

First, let us state an optimization result for the $IS\!D_a$ index on $\mathcal{G}_c(n)$ and $\mathcal{G}(n)$ (see ^[32]).

Proposition 4. Consider $a < 0$ and an integer $n\ge 2$.

$(1)$ The graph that maximizes the $IS\!D_{a}$ index on $\mathcal{G}_c(n)$ or $\mathcal{G}(n)$ is unique and given by the complete graph $K_n$.

$(2)$ If a graph minimizes the $IS\!D_{a}$ index on $\mathcal{G}_c(n)$, then it is a tree.

$(3)$ If $n$ is even, then the graph that minimizes the $IS\!D_{a}$ index on $\mathcal{G}(n)$ is unique and given by the union of $n/2$ paths $P_2$. If $n$ is odd, then the graph that minimizes the $IS\!D_{a}$ index on $\mathcal{G}(n)$ is unique and given by the union of $(n-3)/2$ paths $P_2$ with a path $P_3$.

Proof. Let $G$ be a graph with $n$ vertices, $m$ edges and minimum degree $\delta$.

Items (1) and (2) follow directly from Proposition 3.

For the proof of item (3), we first assume that $n$ is even. For any graph $G\in \mathcal{G}(n)$ Lemma 2 gives

and the equality is attained if, and only if, $\{d_u, d_v\} = \{1\}$ for each $uv\in E(G)$, i.e., $G$ is the union of $n/2$ path graphs $P_2$.

Now, we assume that $n$ is odd. If $d_u = 1$ for each $u\in V(G)$, handshaking lemma gives $2m = n$, a contradiction. So, there exists $w\in V(G)$, such that $d_w\ge2$. Let $N(w)$ be the set of neighbors of the vertex $w$, from Lemma 2, we obtain

and the equality is attained if, and only if, $d_u = 1$ for each $u\in V(G)\setminus w$ and $d_w = 2$. Hence, $G$ is the union of $(n-3)/2$ path graphs $P_2$ and a path graph $P_3$.

Proposition 4 allows to obtain the following inequalities.

Theorem 5. Consider a graph $G$ with $n$ vertices and a negative constant $a$.

$(1)$ Then,

and equality holds if, and only if, $G$ is the complete graph $K_n$.

$(2)$ If $n$ is even, then

and equality holds if, and only if, $G$ is the union of $n/2$ path graphs $P_2$.

$(3)$ If $n$ is odd, then

and equality holds if, and only if, $G$ is the union of a path graph $P_3$ and $(n-3)/2$ path graphs $P_2$.

Proof. Proposition 4 gives

This argument gives that the bound is attained if, and only if, $G$ is the complete graph $K_n$. Hence, item $(1)$ holds.

Suppose $G$ has minimum degree $\delta$. If $n$ is even, handshaking lemma gives $2m\ge n \delta \ge n$, using this and the proof of Proposition 4, we have

and equality holds if, and only if, $G$ is the union of $n/2$ path graphs $P_2$. This gives item (2).

If $n$ is odd, handshaking lemma gives $2m\ge (n-1) \delta +2 \ge n+1$, using this and the proof of Proposition 4, we have

and equality holds if, and only if, $G$ is the union of a path graph $P_3$ and $(n-3)/2$ path graphs $P_2$. This gives item (3).

Fix positive integers $1\le \delta < n$. Let $K_n^\delta$ be the $n$-vertex graph with minimum and maximum degrees $\delta$ and $n-1$, respectively, obtained from $K_{n-1}$ (the complete graph with $n-1$ vertices) and an additional vertex $w$, as follows: If we fix $\delta$ vertices $v_1, \dots, v_\delta \in V(K_{n-1})$, then the vertices of $K_n^\delta$ are $w$ and the vertices of $K_{n-1}$, and the edges of $K_n^\delta$ are $\{v_1w, \dots, v_\delta w\}$ and the edges of $K_{n-1}$.

We consider now the optimization problem for the $IS\!D_a$ index on $\mathcal{H}_c(n, \delta)$ and $\mathcal{H}(n, \delta)$.

Theorem 6. Consider $a < 0$ and integers $1\le \delta < n$.

$(1)$ Then, the graph in $\mathcal{H}_c(n, \delta)$ that maximizes the $IS\!D_{a}$ index is unique and given by $K_n^\delta$.

$(2)$ If $\delta \ge 2$ and $n \delta$ is even, then all the graphs in $\mathcal{H}_c(n, \delta)$ that minimize $IS\!D_{a}$ are the connected $\delta$-regular graphs.

$(3)$ If $\delta \ge 2$ and $n \delta$ is odd, then all the graphs in $\mathcal{H}_c(n, \delta)$ that minimize $IS\!D_{a}$ are the connected $(\delta+1, \delta)$-pseudo-regular graphs.

Proof. Given a graph $G \in \mathcal{H}_c(n, \delta)\setminus \{K_n^\delta\}$, fix any vertex $u \in V(G)$ with $d_u = \delta$. Since

Proposition 3 gives $IS\!D_a(K_n^\delta) > IS\!D_a(G)$. This proves item $(1)$.

Handshaking lemma gives $2m \ge n \delta$.

Since $d_u \ge \delta$ for every $u \in V(G)$, Lemma 2 gives

and the bound is attained if, and only if, $\delta = d_u$ for all $u \in V(G)$.

If $\delta n$ is even, then Lemma 1 gives that there is a connected $\delta$-regular graph with $n$ vertices. Hence, the unique graphs in $\mathcal{H}_c(n, \delta)$ that minimize the $IS\!D_{a}$ index are the connected $\delta$-regular graphs.

If $\delta n$ is odd, then handshaking lemma gives that there is no regular graph. Hence, there exists a vertex $w$ with $d_w \ge \delta+1$. Since $d_u \ge \delta$ for every $u \in V(G)$, handshaking lemma gives $2m \ge (n-1) \delta+\delta+1 = n \delta+1$. Lemma 2 gives

and the bound is attained if, and only if, $d_u = \delta$ for all $u \in V(G)\setminus \{w\}$, and $d_w = \delta+1$. Lemma 1 gives that there is a connected $(\delta+1, \delta)$-pseudo-regular graph with $n$ vertices. Therefore, the unique graphs in $\mathcal{H}_c(n, \delta)$ that minimize the $IS\!D_{a}$ index are the connected $(\delta+1, \delta)$-pseudo-regular graphs.

Remark 1. If we replace $\mathcal{H}_c(n, \delta)$ with $\mathcal{H}(n, \delta)$ everywhere in the statement of Theorem 6, then the argument in its proof gives that the same conclusions hold if we remove everywhere the word "connected".

Theorem 6 and Remark 1 have the following consequence.

Proposition 7. Consider a graph $G$ with minimum degree $\delta$ and $n$ vertices, and a negative constant $a$.

$(1)$ Then,

and the bound is attained if, and only if, $G$ is isomorphic to $K_n^\delta$.

$(2)$ If $\delta \ge 2$ and $\delta n$ is even, then

and the bound is attained if, and only if, $G$ is $\delta$-regular.

$(3)$ If $\delta \ge 2$ and $n \delta$ is odd, then

and the bound is attained if, and only if, $G$ is $(\delta+1, \delta)$-pseudo-regular.

Let us deal with the optimization problem for the $IS\!D_a$ index on $\mathcal{I}_c(n, \Delta)$.

Theorem 8. Consider $a < 0$ and integers $2\le \Delta < n$.

$(1)$ If $n \Delta$ is even, then all the graphs that maximize $IS\!D_{a}$ on $\mathcal{I}_c(n, \Delta)$ are the connected $\Delta$-regular graphs.

$(2)$ If $n \Delta$ is odd, then all the graphs that maximize $IS\!D_{a}$ on $\mathcal{I}_c(n, \Delta)$ are the connected $(\Delta, \Delta-1)$-quasi-regular graphs.

$(3)$ If a graph minimizes $IS\!D_{a}$ on $\mathcal{I}_c(n, \Delta)$, then it is a tree.

Proof. Handshaking lemma gives $2m \le n \Delta$. Since $d_u \le \Delta$ for every $u \in V(G)$, Lemma 2 gives

and the bound is attained if, and only if, $\Delta = d_u$ for all $u \in V(G)$.

If $n \Delta$ is even, then Lemma 1 gives that there is a connected $\Delta$-regular graph with $n$ vertices. Hence, the unique graphs in $\mathcal{I}_c(n, \Delta)$ that maximize the $IS\!D_{a}$ index are the connected $\Delta$-regular graphs.

If $n \Delta$ is odd, then handshaking lemma gives that there is no regular graph in $\mathcal{I}_c(n, \Delta)$. Let $G \in \mathcal{I}_c(n, \Delta)$. Hence, there exists a vertex $w$ with $d_w \le \Delta-1$. Then, $2m \le \Delta(n-1)+ \Delta-1 = \Delta n-1$. Lemma 2 gives

and the bound is attained if, and only if, $d_u = \Delta$ for all $u \in V(G)\setminus \{w\}$, and $d_w = \Delta-1$. Lemma 1 gives that there is a connected $(\Delta, \Delta-1)$-quasi-regular graph with $n$ vertices. Therefore, the unique graphs in $\mathcal{I}_c(n, \delta)$ that maximize the $IS\!D_{a}$ index are the connected $(\Delta, \Delta-1)$-quasi-regular graphs.

Given any graph $G \in \mathcal{I}_c(n, \Delta)$ which is not a tree, fix any vertex $u \in V(G)$ with $d_u = \Delta$. Since $G$ is not a tree, there exists a cycle $C$ in $G$. Since $C$ has at least three edges, there exists $vw \in E(G)\cap C$, such that $u \notin \{v, w\}$. Since $vw$ is contained in a cycle of $G$, then $G\setminus \{vw\}$ is a connected graph. Thus, $G\setminus \{vw\} \in \mathcal{I}_c(n, \Delta)$ and Proposition 3 gives $IS\!D_a(G) > IS\!D_a(G\setminus \{vw\})$. By iterating this argument, we obtain that if a graph minimizes the $IS\!D_{a}$ index on $\mathcal{I}_c(n, \Delta)$, then it is a tree.

The following result deals with the optimization problem for the $IS\!D_a$ index on $\mathcal{I}(n, \Delta)$.

Theorem 9. Consider $a < 0$ and integers $2\le \Delta < n$.

$(1)$ If $n \Delta$ is even, then all the graphs that maximize the $IS\!D_{a}$ index on $\mathcal{I}(n, \Delta)$ are the $\Delta$-regular graphs.

$(2)$ If $n \Delta$ is odd, then all the graphs that maximize the $IS\!D_{a}$ index on $\mathcal{I}(n, \Delta)$ are the $(\Delta, \Delta-1)$-quasi-regular graphs.

$(3)$ If $n- \Delta$ is odd, then the graph that minimizes the $IS\!D_{a}$ index on $\mathcal{I}(n, \Delta)$ is unique and given by the union of the star graph $S_{ \Delta+1}$ and $(n- \Delta-1)/2$ path graphs $P_2$.

$(4)$ If $n = \Delta+2$, then the graph that minimizes the $IS\!D_{a}$ index on $\mathcal{I}(n, \Delta)$ is unique and given by the star graph $S_{ \Delta+1}$ with an additional edge attached to a vertex of degree $1$ in $S_{ \Delta+1}$.

$(5)$ If $n \ge \Delta+4$ and $n- \Delta$ is even, then the graph that minimizes the $IS\!D_{a}$ index on $\mathcal{I}(n, \Delta)$ is unique and given by the union of the star graph $S_{ \Delta+1}$, $(n- \Delta-4)/2$ path graphs $P_2$ and a path graph $P_3$.

Proof. The argument in Theorem 8 gives directly items $(1)$ and $(2)$.

Let $G \in \mathcal{I}(n, \Delta)$ and $w \in V(G)$ a vertex with $d_w = \Delta$.

Assume first that $n- \Delta$ is odd. Handshaking lemma gives $2m \ge n-1+ \Delta$. Note that $n-1+ \Delta = n- \Delta + 2 \Delta-1$ is even. Lemma 2 gives

and the bound is attained if, and only if, $1 = d_u$ for all $u \in V(G) \setminus \{w\}$, i.e., $G$ is the union of the star graph $S_{ \Delta+1}$ and $(n- \Delta-1)/2$ path graphs $P_2$.

Assume now that $n = \Delta+2$. Let $z \in V(G) \setminus N(w)$ be the vertex with $V(G) = \{w, z\} \cup N(w)$. Choose $p \in N(z)$; since $z \notin N(w)$, we have $p \in N(w)$ and so, $d_p \ge 2$. Handshaking lemma gives $2m \ge (n-2) + \Delta +2 = n + \Delta$. Lemma 2 gives

and the bound is attained if, and only if, $1 = d_u$ for all $u \in V(G) \setminus \{w, p\}$ and $d_p = 2$, i.e., $G$ is the star graph $S_{ \Delta+1}$ with an additional edge attached to a vertex of degree $1$ in $S_{ \Delta+1}$.

Assume that $n \ge \Delta+4$ and $n- \Delta$ is even. If $d_u = 1$ for every $u \in V(G) \setminus \{w\}$, then handshaking lemma gives $2m = n-1+ \Delta$, a contradiction since $n-1+ \Delta = n- \Delta+2 \Delta-1$ is odd. Thus, there exists a vertex $p \in V(G) \setminus \{w\}$ with $d_p \ge 2$. Handshaking lemma gives $2m \ge (n-2) +2 + \Delta = n + \Delta$.

If $p \notin N(w)$, then Lemma 2 gives

and the bound is attained if, and only if, $d_u = 1$ for all $u \in V(G)\setminus \{w, p\}$, and $d_p = 2$, i.e., $G$ is the union of the star graph $S_{ \Delta+1}$, $(n- \Delta-4)/2$ path graphs $P_2$ and a path graph $P_3$.

If $p \in N(w)$, then

Hence, in order to finish the proof of item $(5)$, it suffices to show that

We have

and so, $(5)$ holds.

Remark 2. Note that the case $\Delta = 1$ in Theorem 9 is trivial: if $\Delta = 1$, then $G$ is a union of isolated edges.

Also, we can state the following inequalities.

Theorem 10. Consider a graph $G$ with maximum degree $\Delta$ and $n$ vertices, and a negative constant $a$.

$(1)$ If $n \Delta$ is even, then

and the bound is attained if, and only if, $G$ is a regular graph.

$(2)$ If $n \Delta$ is odd, then

and the bound is attained if, and only if, $G$ is a $(\Delta, \Delta-1)$-quasi-regular graph.

$(3)$ If $n- \Delta$ is odd, then

and the bound is attained if, and only if, $G$ is the union of the star graph $S_{ \Delta+1}$ and $(n- \Delta-1)/2$ path graphs $P_2$.

$(4)$ If $n = \Delta+2$, then

and the bound is attained if, and only if, $G$ is the star graph $S_{ \Delta+1}$ with an additional edge attached to a vertex of degree $1$ in $S_{ \Delta+1}$.

$(5)$ If $n \ge \Delta+4$ and $n- \Delta$ is even, then

and the bound is attained if, and only if, $G$ is the union of the star graph $S_{ \Delta+1}$, $(n- \Delta-4)/2$ path graphs $P_2$ and a path graph $P_3$.

Proof. The argument in the proof of Theorem 8 gives items $(1)$ and $(2)$, since the variable inverse sum deg index of a regular graph is

and the $IS\!D_a$ index of a $(\Delta, \Delta-1)$-quasi-regular graph is

The argument in the proof of Theorem 9 gives directly items $(3)$–$(5)$.

3.   QSPR study of $IS\!D_a$ on polyaromatic hydrocarbons

The variable inverse sum deg index $IS\!D_{-1.950}$ was selected in ^[33] as a significant predictor of standard enthalpy of formation for octane isomers. In this section, we will test the predictive power of the $IS\!D_a$ index using experimental data on three physicochemical properties of 82 polyaromatic hydrocarbons (PAH). The properties studied are the melting point (MP), boiling point (BP) and octanol-water partition coefficient (LogP) (the experimental data were obtained from ^[34]). In order to obtain the values of the $IS\!D_a$ index, we constructed the hydrogen-suppressed graph of each molecule, then we use a program of our own elaboration to compute the index for each value of $a$ analyzed.

We calculated the Pearson's correlation coefficient $r$ between the three analyzed properties and the $IS\!D_a$ index, for values of $a$ in the interval $[-5, 5]$ with a spacing of $0.01$; the results are shown in Figure 1. The dashed red line indicates the value of $a$ that maximizes $r$.

Figure 2 shows the $IS\!Da$ index (for values of $a$ that maximize $r$) vs. the studied properties of PAH. In addition, in Figure 2, we test the following linear regression models (red lines)

Table 1 summarizes the statistical and regression parameters of these models.

4.   Conclusions

Motivated by a paper of Vukičević ^[1], and based on the practical applications found for the variable inverse sum deg index $IS\!D_{a}$, we focus our research on the study of optimal graphs associated with $IS\!D_{a}$, when $a < 0$. In this direction, it is wise to study the extremal properties of $IS\!D_{a}$, when $a < 0$ in general graphs. Specifically, in this paper, we characterize the graphs with extremal values in the following significant classes of graphs with a fixed number of vertices:

● graphs with a fixed minimum degree;

● connected graphs with a fixed minimum degree;

● graphs with a fixed maximum degree;

● connected graphs with a fixed maximum degree.

From the QSPR study performed on polyaromatic hydrocarbons, it can be concluded that the $IS\!D_a$ index presents a strong correlation with the boiling point and octanol-water partition coefficient properties, with maximum values of $r$ higher than $0.98$ and $0.94$, respectively. Further, the melting point property presents some correlation with the $IS\!D_a$ index with maximum value of $r$ close to $0.85$.

For future research, we suggests:

● To study the extreme problems for the $IS\!D_a$ index for values of $a > 0$.

● To consider the problem of finding which tree/trees with $n$ vertices (with a fixed maximum degree or not) minimize the index $IS\!D_a$ ($a < 0$).

● To analyze the behavior of the $IS\!D_{a}$ index in other important families of graphs, such as graph products and graph operators.

● To explore the mathematical properties and possible applications of the exponential extension of the $IS\!D_a$ index.

● To study the predictive power of this index on other physicochemical properties of PAH, and on other classes of molecules.

Acknowledgments

The authors were supported by a grant from Agencia Estatal de Investigación (PID2019-106433GB-I00 / AEI / 10.13039/501100011033), Spain.

Conflict of interest

The authors declare there are no conflicts of interest.