I.D {streamDAG}

Generalized DAG indices

Description

Calculates global generalized topopological indices for a digraph

Usage

I.D(G, mode = "gen.rand", alpha = -1/2, mult = FALSE, degrees = "out.in")

Arguments

G

Graph object of class. See graph_from_literal.

mode

One of "gen.rand","gen.sum.con", "ABC", "GA", "harm", "aug.rand".

alpha

Exponent value for forms of omega with alpha exponent.

mult

Logical if TRUE use experimental multiplicative measures.

degrees

Degree designations for the arc \vec{uv}. One of "out.in", "out.out", "in.out", "in.in". See Details below. The default designation, "out.in", is strongly reccomended for stream DAGs.

Details

For an arc a={\overrightarrow{uv}}, a \in A, we denote the out degree of u as d_u^+, and the in degree of v as d_v^-. Now let I(D) represent a generalized topopological index for a digraph, D (cf. Deng et al., 2021) that depends on d_u^+ and d_v^-:

I(D)=1/2 \sum_{uv \in A}\omega(d_u^+,d_v^-)

Six basic configurations for I(D) can be recognized:

1. If \omega(x,y)=(xy)^\alpha, for \alpha \neq 0, then I(D) is the general directed Randic index (Kincaid et al., 2016) for D. Specific variants include the Randic index (\alpha =-1/2), the second Zagreb index (\alpha =1) and the second modified Zagreb index (\alpha =-1) (Anthony & Marr, 2021).

2. If \omega(x,y)=(x+y)^\alpha, then I(D) is the general sum-connectivity index for D (Deng et al., 2021). Further, if \omega(x,y)=2(x+y)^\alpha, then I(D) is the sum connectivity (Zhou & Trinajstic, 2009), and the directed first Zagreb index (Anthony & Marr, 2021) for \alpha =-1/2 and \alpha=1, respectively .

3. If \omega(x,y)=\sqrt{((x+y-2)/xy)}, then I(D) is the atom bond connectivity of D (Estrada et al., 1998).

4. If \omega(x,y)=\sqrt{xy}/(1/2(x+y)), then I(D) is the geometric-arithmetic index for D (Vukicevic & Furtula, 2009).

5. If \omega(x,y)=2/(x+y), then I(D) is the harmonic index of D (Favaron et al., 1993).

6. If \omega(x,y)=\left(\frac{xy}{x+y-2}\right)^3, then I(D) is the augmented Randic index of D (Furtula et al. 2010). This index is not reccomended for stream DAGs as it will contained undefined terms for any network with unbranched paths.

More options are possible under the generalization of Kincaid (1996). Specifically, for an arc a=\vec{uv},a\in A, let \gamma,\tau\in-,+ index the degree type: -= in, +=out. Then, four combinations of d_u^\gamma, d_v^\tau can occur, resulting in four different versions of each I(D) metric described above. These combinations are: d_u^+,d_v^- (as shown above), d_u^+,d_v^+, d_u^-,d_v^-, and d_u^-,d_v^+. The default d_u^+,d_v^- is strongly reccomended for stream DAGs over other variants.

Value

Index values for a DAG

Author(s)

Ken Aho, Gabor Csardi wrote degree

References

Anthony, B. M., & Marr, A. M. (2021). Directed zagreb indices. Graphs and Combinatorial Optimization: From Theory to Applications: CTW 2020 Proceedings, 181-193.

Deng, H., Yang, J., Tang, Z., Yang, J., & You, M. (2021). On the vertex-degree based invariants of digraphs. arXiv Preprint arXiv:2104.14742.

Estrada, E., Torres, L., Rodriguez, L., & Gutman, I. (1998). An atom-bond connectivity index: Modelling the enthalpy of formation of alkanes. NISCAIR-CSIR, India.

Favaron, O., Maheo, M., & Sacle, J.-F. (1993). Some eigenvalue properties in graphs (conjectures of graffitii). Discrete Mathematics, 111(1-3), 197-220.

Furtula, B., Graovac, A., & Vukicevic, D. (2010). Augmented Zagreb index. Journal of Mathematical Chemistry, 48(2), 370-380.

Kincaid, R. K., Kunkler, S. J., Lamar, M. D., & Phillips, D. J. (2016). Algorithms and complexity results for finding graphs with extremal Randic index. Networks, 67(4), 338-347.

Vukicevic, D., & Furtula, B. (2009). Topological index based on the ratios of geometrical and arithmetical means of end-vertex degrees of edges. Journal of Mathematical Chemistry, 46(4), 1369-1376.

Zhou, B., & Trinajstic, N. (2009). On a novel connectivity index. Journal of Mathematical Chemistry, 46(4), 1252-1270.

See Also

degree

Examples

network_a <- graph_from_literal(a --+ b, c --+ d, d --+ e, b --+ e, 
e --+ j, j --+ m, f --+ g, g --+ i, h --+ i, i --+ k, k --+ l, 
l --+ m, m --+ n,  n --+ o)
I.D(network_a)