The Euler-Sombor index \(EU\) is a vertex-degree-based graph invariant, defined as the sum over all pairs of adjacent vertices \(u,v\) of the underlying graph, of the terms \(\sqrt{d_u^2+d_v^2+d_u\,d_v}\), where \(d_u\) and \(d_v\) are the degrees of the vertices \(u\) and \(v\), respectively. For a real number \(\lambda\), a variable version of \(EU\) is constructed, denoted by \(EU(\lambda)\), defined via \(\sqrt{d_u^2+d_v^2+\lambda\,d_u\,d_v}\). Its special cases for \(\lambda=2,\,-2,\,0\), and 1 are, respectively, the first Zagreb, Albertson, Sombor, and the ordinary Euler-Sombor indices. The basic properties of \(EU(\lambda)\) are determined, including a method for its approximate calculation and bounds in terms of minimum degree, maximum degree, order and size for several graph products. It is shown how to find values of \(\lambda\) for which \(EU(\lambda)\) is optimal with regard to predicting molecular properties.
The Sombor index \(SO\) is a recently discovered vertex-degree-based graph invariant [1, 2], defined as \[\label{3} SO = SO(G)=\sum_{uv\in \mathbf E(G)} \sqrt{d_u^2 + d_v^2}\,. \tag{1}\]
Initially, it was aimed at solving chemical problems [3] but in the meantime it found peculiar applications is various other fields of science and technology, see e.g. [4– 7]. The mathematical properties of \(SO(G)\) are nowadays studied in due detail [8]. In [9] an elliptic geometric model for Sombor-type indices was proposed. Within this model, the perimeter of the considered ellipse could be (approximately) calculated by means of Euler’s formula [10] \(\pi\,\sqrt{2(r_1+r_2)}\), where \(r_1\) and \(r_2\) are the semi-major and semi-minor axes of the ellipse. In [9] it was shown that \[r_1 = \sqrt{d_u^2+d_v^2} \hspace{10mm} \mbox{and} \hspace{10mm} r_2 = \frac{1}{\sqrt{2}}\,(d_u+d_v),\] which implies that the perimeter is \(\sqrt{3}\pi\,\sqrt{d_u^2+d_v^2+ \frac{2}{3}\,d_u\,d_v}\). This was the motivation to introduce a new Sombor-type graph invariant [11], defined as \[\label{4} EU = EU(G)=\sum_{uv\in \mathbf E(G)} \sqrt{d_u^2 + d_v^2 + d_u\,d_v}, \tag{2}\] and named Euler-Sombor index, see also [12– 14].
In the present paper we examine a variable version of the Euler-Sombor index, defined as \[\label{5} EU(\lambda) = EU(\lambda,G)=\sum_{uv\in \mathbf E(G)} \sqrt{d_u^2 + d_v^2 + \lambda\,d_u\,d_v}, \tag{3}\] where \(\lambda\) is a real number.
In the above formulas, \(G\) stands for a simple graph with vertex set \(\mathbf V(G)\) and edge set \(\mathbf E(G)\). If the vertices \(u,v \in \mathbf V(G)\) are adjacent, then the edge connecting them is denoted by \(uv\). The degree of a vertex \(u \in \mathbf V(G)\) is denoted by \(d_u\). The summations in the above formulas go over all edges \(uv\) of the underlying graph \(G\). For other graph-theoretical notation and terminology, we refer to [15, 16]. For details of chemical graph theory, especially for applications of topological indices see [17, 18].
Today, in the mathematical and chemical literature there is a legion of vertex-degree-based graph invariants [19– 22]. Of these, for the present considerations we need the following two.
The oldest vertex-degree-based molecular structure descriptor is the first Zagreb index, [23], defined as \[\label{1} M_1 = M_1(G)=\sum_{uv\in \mathbf E(G)} \big[ d_u + d_v \big]\,. \tag{4}\]
The oldest graph-theoretical irregularity measure is the Albertson index [24] \[\label{2} Alb = Alb(G)=\sum_{uv\in \mathbf E(G)} \big| d_u – d_v \big|\,. \tag{5}\]
For details on the first Zagreb and Albertson irregularity indices see [25, 26] and [27, 28], respectively.
At the first glance, formulas (1)–(5) appear to have quite different algebraic forms. Yet, they may be related using the following results that are obtained by direct calculation, taking into account Eqs. (1)–(5).
Proposition 1. Let \(\lambda\) be a real number and let \(EU(\lambda,G)\) be the corresponding variable Euler-Sombor index. Then \[EU(2,G) = M_1(G) , \label{6} \ \tag{6}\] \[EU(-2,G) = Alb(G) , \label{7} \ \tag{7}\] \[EU(0,G) = SO(G) , \label{8} \ \tag{8}\] \[EU(1,G) = EU(G) . \label{9} \tag{9}\]
Equalities (6)–(9) hold for any (simple) graph \(G\).
The relations stated in Proposition 1 may be considered as the basic mathematical property of the variable Euler-Sombor index.
Proposition 2. In the general case, \(EU(\lambda)\) is well-defined (i.e., real-valued) only for \(\lambda \geq -2\).
Proof. In formula (3), the term \(d_u^2 + d_v^2 + \lambda\,d_u\,d_v\) must always be positive-valued or zero. Noting that \(d_u^2 + d_v^2 + \lambda\,d_u\,d_v = (d_u-d_v)^2 + (\lambda+2)d_u\,d_v\), and that in some graphs the case \(d_u=d_v\) may happen, we see that it must be \(\lambda+2 \geq 0\). ◻
Although for some chosen value of the variable \(\lambda\), the actual value of the function \(EU(\lambda)\) can be directly calculated by means of Eq. (3), a much easier way to do this would be by using the relations (6)–(9). Because the properties of the first Zagreb, Albertson irregularity, and Sombor indices are best understood and most detailed investigated, see [25, 26], [27, 28], and [8], respectively, we use the formulas (6)–(8) to obtain a second-degree polynomial approximation. By setting \[EU(\lambda) \approx a\,\lambda^2 + b\,\lambda + c,\] and adjusting that for \(\lambda=-2, 0\), and +2, \(a\,\lambda^2 + b\,\lambda + c\) be equal to \(Alb\), \(SO\), and \(M_1\), respectively, we obtain
\[\begin{aligned} a = & \frac{1}{8}(M_1+Alb-2\,SO), \\ b = & \frac{1}{4}(M_1 – Alb), \\ c = & SO, \end{aligned}\] i.e., \[\label{10} EU(\lambda) \approx \frac{\lambda^2}{8}\,(M_1+Alb-2\,SO) + \frac{\lambda}{4}\,(M_1 – Alb)+ SO. \tag{10}\]
The approximation (10) is best applicable for \(\lambda \in [-2,+2]\). Its quality is seen from Figure 1.
In this section, we assume that the graphs considered are simple, finite, undirected and connected. For details on graph products see [29].
The corona product of graphs \(G\) and \(H\) is the graph denoted by \(G\bigodot H\), obtained by taking one copy of \(G\) and \(n_G\) copies of \(H\), and joining the vertex \(u\) that is on \(i\)-th position in \(G\) to every vertex in \(i\)-th copy of \(H\). The order and size of \(G\bigodot H\) are \(n_G(1+n_H)\) and \(m_G+n_Gm_H+n_Gn_H\), respectively.
The degree of a vertex \(u \in \mathbf V(G\bigodot H)\) is given by \[\label{11} d_{G\bigodot H} (u)= \begin{cases} d_G(u)+n_H, & if u\in \mathbf V(G),\\ d_H(u)+1, & if u\in \mathbf V(H). \end{cases} \tag{11}\]
Theorem 1. Let \(G\) and \(H\) be graphs with maximum degrees \(\Delta_G\,\Delta_H\) and minimum degrees \(\delta_G,\delta_H\), respectively. Then, \(\mu_1\leq EU(\lambda,G\bigodot H)\leq\mu_2\), where \[\begin{aligned} \mu_1 = m_G\sqrt{\lambda+2}(\delta_G+n_H)+m_H\sqrt{\lambda+2}(\delta_H+1)+ n_G\,n_H\sqrt{\lambda(\delta_G+n_H)(\delta_H+1)}, \end{aligned}\] and \[\begin{aligned} \mu_2 = m_G\sqrt{\lambda+2}(\Delta_G+n_H)+m_H\sqrt{\lambda+2}(\Delta_H+1) + n_G\,n_H\sqrt{\lambda(\Delta_G+n_H)(\Delta_H+1)}\,. \end{aligned}\]
Equality holds if and only if \(G\) and \(H\) are regular graphs.
Proof. By using the definitions of variable Euler-Sombor index and (11), we obtain \[\begin{aligned} EU(\lambda,G\bigodot H) = & \sum_{uv \in \mathbf E(G)}\sqrt{(d_G(u)+n_H)^2+(d_G(v)+n_H)^2+\lambda(d_G(u)+n_H)(d_G(v)+n_H)}\\ & + n_G\sum_{uv \in \mathbf E(H)}\sqrt{(d_H(u)+1)^2+(d_H(v)+1)^2+\lambda(d_H(u)+1)(d_H(v)+1)}\\ & + \sum_{u\in \mathbf V(G)}\,\sum_{u\in \mathbf V(H)} \sqrt{(d_G(u)\!+\!n_H)^2\!+\!(d_G(v)\!+\!1)^2\!+\!\lambda(d_G(u)\!+\!n_H)(d_G(v)+1)}\\ \leq &m_G\sqrt{\lambda\!+\!2}\,(\Delta_G\!+\!n_H)\!+\!m_H\sqrt{\lambda\!+\!2}\,(\Delta_H\!+\!1) \!+\!n_G\,n_H\sqrt{\lambda(\Delta_G\!+\!n_H)(\Delta_H\!+\!1)}\,. \end{aligned}\]
The lower bound is obtained in an analogous manner. ◻
The Cartesian product of \(G\) and \(H\) is the graph denoted by \(G\times H\), with vertex set \(\mathbf V(G)\times \mathbf V(H)\) and two vertices \(u=(u_1,v_1)\) and \(v=(u_2,v_2)\) being adjacent in \(G\times H\) whenever \(u_1=u_2\) and \(v_1\) and \(v_2\) are adjacent in \(H\) or \(v_1=v_2\) and \(u_1\) and \(u_2\) are adjacent in \(G\). The size of the Cartesian product of the graphs \(G\) and \(H\) is \(m_Gn_H+n_Gm_H\).
The degree of a vertex \((u,v) \in \mathbf V(G\times H)\) is \[\label{12} d_{G\times H}(u,v)=d_G(u)+d_H(v)\,. \tag{12}\]
Theorem 2. Let \(G\) and \(H\) be graphs with maximum degrees \(\Delta_G\),\(\Delta_H\) and minimum degrees \(\delta_G\),\(\delta_H\), respectively. Then, \[\sqrt{\lambda+2}(\delta_G+\delta_H)m_{G\times H}\leq EU(\lambda,G\times H)\leq \sqrt{\lambda+2}(\Delta_G+\Delta_H)m_{G\times H}\,.\]
Equality on both sides holds if and only if \(G\) and \(H\) are regular graphs.
Proof. By using the definitions of variable Sombor-Euler index and (12), and setting \[\begin{aligned} X_1 = (d_G(u_1)+d_H(v_1))^2+(d_G(u_1)+d_H(v_2))^2 + \lambda(d_G(u_1)+d_H(v_1))(d_G(u_1)+d_H(v_2)), \end{aligned}\] and \[\begin{aligned} X_2 = (d_G(u_1)+d_H(v_1))^2+(d_G(u_2)+d_H(v_1))^2 + \lambda(d_G(u_1)+d_H(v_1))(d_G(u_2)+d_H(v_1)), \end{aligned}\] we obtain \[\begin{aligned} EU(\lambda,G\times H) = & \sum_{u_1 \in \mathbf V(G)}\,\sum_{v_1v_2 \in \mathbf E(H)} \sqrt{X_1} + \sum_{v_1 \in \mathbf V(G)}\, \sum_{u_1u_2 \in \mathbf E(G)} \sqrt{X_2} \\ \leq &\sqrt{\lambda+2}(\Delta_G+\Delta_H)(m_G\,n_H+n_G\,m_H) \\ = & \sqrt{\lambda+2} (\Delta_G+\Delta_H)m_{G\times H}\,. \end{aligned}\]
The lower bound is obtained in an analogous manner. ◻
The lexicographic product of \(G\) and \(H\) is the graph denoted by \(G[H]\), whose vertex set is \(\mathbf V(G)\times \mathbf V(H)\), and two vertices \(u=(u_1,u_2)\) and \(v=(v_1,v_2)\) are adjacent in \(G[H]\) whenever \(u_1v_1 \in \mathbf E(G)\) or \(u_1=v_1\) and \(u_2v_2 \in \mathbf E(H)\). The size of \(G[H]\) is \(m_G\,n_H^2+n_G\,m_H\).
The degree of a vertex \((u,v) \in \mathbf V(G[H])\) is \[ d_{G[H]}(u,v)=n_Hd_G(u)+d_H(v)\,. \tag{13}\]
Theorem 3. Let \(G\) and \(H\) be graphs with maximum degrees \(\Delta_G,\Delta_H\) and minimum degrees \(\delta_G,\delta_H\), respectively. Then, \[\sqrt{\lambda+2}(n_H\delta_G+\delta_H)m_{G[H]}\leq EU(\lambda,G[H])\leq \sqrt{\lambda+2}(n_H\Delta_G+\Delta_H)m_{G[H]}\,.\]
The equality on both sides holds if and only if \(G\) and \(H\) are regular graphs.
Proof. By using the definitions of variable Euler-Sombor index and (13), and setting \[\begin{aligned} Y_1 = (n_H\,d_G(u_1)+d_H(v_1))^2+(n_H\,d_G(u_1)+d_H(v_2))^2 + \lambda(n_H\,d_G(u_1)+d_H(v_1))(n_H\,d_G(u_1)+d_H(v_2)), \end{aligned}\] and \[\begin{aligned} Y_2 = (n_Hd_G(u_1)+d_H(v_1))^2+(n_Hd_G(u_2)+d_H(v_2))^2 + \lambda(n_Hd_G(u_1)+d_H(v_1))(n_Hd_G(u_2)+d_H(v_2)), \end{aligned}\] we obtain \[\begin{aligned} EU(\lambda,G[H])&=\sum_{u_1 \in \mathbf V(G)}\sum_{v_1v_2 \in \mathbf E(H)} \sqrt{Y_1} +\sum_{v_1 \in \mathbf V(H)}\sum_{v_2 \in \mathbf V(H)}\sum_{u_1u_2 \in \mathbf E(G)} \sqrt{Y_2} \\ &\leq \sqrt{\lambda+2}(n_H\,\Delta_G+\Delta_H)(m_G\,n_H^2+n_G\,m_H)\\ &=\sqrt{\lambda+2} (n_H\,\Delta_G+\Delta_H)m_{G[H]}\,. \end{aligned}\] ◻
The strong product of \(G\) and \(H\) is the graph denoted by \(G \boxtimes H\), whose vertex set is \(\mathbf V(G)\times \mathbf V(H)\), and two vertices \(u=(u_1,u_2)\) and \(v=(v_1,v_2)\) are adjacent in \(G \boxtimes H\) whenever \(u_1=u_2\) and \(v_1\) and \(v_2\) are adjacent in \(H\) or \(v_1=v_2\) and or \(u_1\) and \(u_2\) are adjacent in \(G\) or \(u_1\) and \(u_2\) are adjacent in \(G\) and \(v_1\) and \(v_2\) are adjacent in \(H\).
The degree of a vertex \((u,v) \in V(G \boxtimes H)\) is \[\label{14} d_{G \boxtimes H}(u,v)=d_G(u)+d_H(v)+d_G(u)\,d_H(v)\,. \tag{14}\]
Theorem 4. Let \(G\) and \(H\) be graphs with maximum degrees \(\Delta_G,\Delta_H\) and minimum degrees \(\delta_G,\delta_H\), respectively. Then, \[\begin{aligned} \sqrt{\lambda+2}(\delta_G+\delta_H+(\delta_G\,\delta_H))m_{G \boxtimes H} \leq EU(\lambda,G \boxtimes H) \leq \sqrt{\lambda+2}(\Delta_G+\Delta_H+(\Delta_G\,\Delta_H))m_{G \boxtimes H}\,. \end{aligned}\]
The equality on both sides holds if and only if \(G\) and \(H\) are regular graphs.
Proof. By taking \[\begin{aligned} A= &(d_G(u_1)+d_H(v_1)+d_G(u_1)\,d_H(v_1))^2,\\ B= &(d_G(u_1)+d_H(v_2)+d_G(u_1)\,d_H(v_2))^2,\\ C= &(d_G(u_2)+d_H(v_1)+d_G(u_2)\,d_H(v_1))^2, \end{aligned}\] and using Eq. (14), we get \[\begin{aligned} EU(\lambda,G \boxtimes H)&=\sum_{u_1 \in \mathbf V(G)}\,\sum_{v_1v_2 \in E(H)}\sqrt{A+B+\lambda(AB)} +\sum_{v_1 \in \mathbf V(H)}\,\sum_{u_1u_2 \in \mathbf E(G)}\sqrt{A+C+\lambda(AC)} \\ & \quad+2\sum_{u_1u_2 \in \mathbf E(G)}\,\sum_{v_1v_2 \in \mathbf E(H)}\sqrt{A+D+\lambda(AD)}\\ &\leq \sqrt{\lambda+2}(\Delta_G+\Delta_H+(\Delta_G\,\Delta_H))(n_G\,m_H+m_G\,n_H+2m_G\,m_H)\\ &=\sqrt{\lambda+2} (\Delta_G+\Delta_H+(\Delta_G\,\Delta_H))m_G\boxtimes H. \end{aligned}\]
The lower bound is obtained analogously. ◻
Within this section, we consider as an example the standard entropy \(S^0\) of the set of 18 isomeric octanes. What immediately comes to the mind is that by using values of \(\lambda\), different from -2,0,1,2, we may improve the correlation between \(S^0\) and \(EU(\lambda)\). This would mean that the structure-dependence of \(S^0\) is modeled by means of some (not necessarily linear) combination of topological indices \(M_1\), \(SO\), \(EU\), and \(Alb\). Unfortunately, as seen from Figure 2, this simple approach is not successful (at least in the case of entropy, at least in the case of octane isomers).
Bearing this difficulty in mind, a multilinear correlation between \(S^0\) and \(EU(\lambda)\) would be necessary to consider. Its general form would be \(EU(\lambda) + \alpha\,EU(\mu)\), with variable \(\lambda\), \(\mu\), and \(\alpha\).
Our preliminary testing indicates that the “optimal” values for the parameters \(\lambda\), \(\mu\), and \(\alpha\) depend very much on the physico-chemical quantity considered, and on the data set used. Therefore, in what follows we only present two characteristic examples.
Example 1. As a first guess, we choose \(\mu=-\lambda\) and, to simplify, set \(\alpha=1\). The results thus obtained are shown in Figure 3.
Example 2. The curve depicted in Figure 3 is symmetric with regard to \(\lambda=0\) because the underlying model was chosen to be symmetric. In order to eliminate such symmetry, we now consider the model \(\mu=-\lambda\) and \(\alpha=1/2\), see Figure 4.
In view of Eq. (10), the model \(EU(\lambda)+(1/2)\,EU(-\lambda)\) yields optimal results at the following peculiar combinations of topological indices: \[\begin{aligned} S^0 \approx & \frac{3 \cdot0.7^2}{16}(M_1+Alb+2\,SO)+ \frac{0.7}{8}(M_1-Alb) + \frac{3}{2}\,SO \\ = & 7 0.179375\,M_1 – 0.004375\,Alb + 1.315250\,SO\,. \end{aligned}\]
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