First General Zagreb Index

• The first general Zagreb index is a degree-power sum invariant for graphs, generalizing classical indices by using a general exponent (α) in its definition.
• It plays a central role in graph theory by offering combinatorial representations via star counts, generating functions, and linear recurrences.
• The index enables the derivation of sharp lower and upper bounds that incorporate degree heterogeneity, with applications ranging from extremal graph theory to chemical-graph modeling.

The first general Zagreb index is a degree-based topological index of a simple graph that extends the classical first Zagreb index by replacing the quadratic degree term with a general power. In the literature represented here, it appears in several equivalent notational forms, most commonly

$Z_{\alpha}(G)=\sum_{v\in V(G)} d(v)^{\alpha},$

or

$M_1^{\alpha}(G)=\sum_{v\in V(G)} d(v)^{\alpha},$

while the variable notation

${}^\lambda M_1=\sum_{i=1}^n d_i^{\,2\lambda}$

corresponds to the parameter choice $\alpha=2\lambda$ (Ilić et al., 2011). The index contains the ordinary first Zagreb index at $\alpha=2$, the handshaking sum at $\alpha=1$, and the $F$-index at $\alpha=3$ (Verma et al., 20 Aug 2025). Its study spans sharp lower and upper bounds, equality characterizations, combinatorial representations through star counts, linear recurrences, extremal behavior on constrained graph classes, exact formulas on lattice networks, and relationships with coindices, spectral bounds, and chemical-graph applications (Bedratyuk et al., 2017).

1. Definition and notational conventions

Let $G=(V,E)$ be a finite simple graph with $n=|V|$, $M_1^{\alpha}(G)=\sum_{v\in V(G)} d(v)^{\alpha},$0, degree sequence $M_1^{\alpha}(G)=\sum_{v\in V(G)} d(v)^{\alpha},$1, and average degree $M_1^{\alpha}(G)=\sum_{v\in V(G)} d(v)^{\alpha},$2. The first general Zagreb index is defined in the surveyed papers as a degree power sum. One widely used convention is

$M_1^{\alpha}(G)=\sum_{v\in V(G)} d(v)^{\alpha},$3

for $M_1^{\alpha}(G)=\sum_{v\in V(G)} d(v)^{\alpha},$4, with the proviso that if $M_1^{\alpha}(G)=\sum_{v\in V(G)} d(v)^{\alpha},$5 then $M_1^{\alpha}(G)=\sum_{v\in V(G)} d(v)^{\alpha},$6 is assumed to avoid division by zero (Verma et al., 20 Aug 2025). Another convention writes the same object as

$M_1^{\alpha}(G)=\sum_{v\in V(G)} d(v)^{\alpha},$7

and explicitly identifies it with the “generalized first Zagreb index,” the “first general Zagreb index,” or the “zeroth-order general Randić index” (Vaidya et al., 2024). In the variable notation of (Ilić et al., 2011),

$M_1^{\alpha}(G)=\sum_{v\in V(G)} d(v)^{\alpha},$8

so that $M_1^{\alpha}(G)=\sum_{v\in V(G)} d(v)^{\alpha},$9.

The specializations are standard across the sources: ${}^\lambda M_1=\sum_{i=1}^n d_i^{\,2\lambda}$0, ${}^\lambda M_1=\sum_{i=1}^n d_i^{\,2\lambda}$1, ${}^\lambda M_1=\sum_{i=1}^n d_i^{\,2\lambda}$2, ${}^\lambda M_1=\sum_{i=1}^n d_i^{\,2\lambda}$3, ${}^\lambda M_1=\sum_{i=1}^n d_i^{\,2\lambda}$4, and ${}^\lambda M_1=\sum_{i=1}^n d_i^{\,2\lambda}$5 (Verma et al., 20 Aug 2025). The parameter domain depends on the framework: the star-sequence formulas and recurrences are developed for integer ${}^\lambda M_1=\sum_{i=1}^n d_i^{\,2\lambda}$6 (Bedratyuk et al., 2017), the convexity-based decomposition results use ${}^\lambda M_1=\sum_{i=1}^n d_i^{\,2\lambda}$7 (Vaidya et al., 2024), and the Hölder-type lower bound in the variable notation is proved under ${}^\lambda M_1=\sum_{i=1}^n d_i^{\,2\lambda}$8, equivalently ${}^\lambda M_1=\sum_{i=1}^n d_i^{\,2\lambda}$9 (Ilić et al., 2011).

Source	Notation	Parameter regime
(Ilić et al., 2011)	$\alpha=2\lambda$0	$\alpha=2\lambda$1 in the lower-bound derivation
(Verma et al., 20 Aug 2025)	$\alpha=2\lambda$2	$\alpha=2\lambda$3; if $\alpha=2\lambda$4, assume $\alpha=2\lambda$5
(Vaidya et al., 2024)	$\alpha=2\lambda$6	$\alpha=2\lambda$7

A related object is the general Zagreb coindex

$\alpha=2\lambda$8

which satisfies

$\alpha=2\lambda$9

for every $\alpha=2$0 (Verma et al., 20 Aug 2025). This identity links the first general Zagreb index to nonedge-based degree sums and is one of the main translation devices from bounds on $\alpha=2$1 to bounds on coindices.

2. Baseline inequalities and the role of regularity

A central structural fact is that the normalized first and second Zagreb indices share the same sharp lower bound in the classical case:

$\alpha=2$2

with equality if and only if $\alpha=2$3 is regular (Ilić et al., 2011). For $\alpha=2$4, this is the Cauchy–Schwarz bound

$\alpha=2$5

obtained from

$\alpha=2$6

and equality holds precisely when all degrees are equal (Ilić et al., 2011).

The same phenomenon extends to the variable first Zagreb index. For $\alpha=2$7,

$\alpha=2$8

with equality if and only if $\alpha=2$9 is regular (Ilić et al., 2011). The proof uses Hölder’s inequality with $\alpha=1$0 and $\alpha=1$1 on $\alpha=1$2 and $\alpha=1$3.

The same paper gives simultaneous sharp upper bounds in terms of the maximum degree $\alpha=1$4:

$\alpha=1$5

and, in the variable setting,

$\alpha=1$6

with equality attained simultaneously if and only if $\alpha=1$7 is regular (Ilić et al., 2011). For the standard index, a refined upper bound involving both $\alpha=1$8 and $\alpha=1$9 is also stated:

$F$0

which implies the corresponding bound for $F$1 (Ilić et al., 2011).

Regularity therefore plays a double role: it is the unique equality case for the common average-degree lower bound, and it is also the unique equality case for the simultaneous maximum-degree upper bounds. For a $F$2-regular graph, $F$3 and

$F$4

so both the lower and upper normalized bounds are attained exactly (Ilić et al., 2011).

3. Refined lower bounds and degree heterogeneity

Beyond the average-degree bound, recent work develops variance-enhanced lower bounds that incorporate selected degrees explicitly. A key result is a refined quadratic inequality for $F$5 real numbers $F$6 with mean $F$7:

$F$8

with equality if and only if all elements except the two distinguished ones are equal (Verma et al., 20 Aug 2025). Substituting $F$9 yields a family of lower bounds for $\alpha=3$0.

For any distinct $\alpha=3$1 and any $\alpha=3$2,

$\alpha=3$3

The paper also proves analogous three-degree and four-degree bounds by repeating the same lemma on subsets of size $\alpha=3$4 and $\alpha=3$5 (Verma et al., 20 Aug 2025). Equality is characterized by degree-multiplicity configurations in which all transformed degrees except the distinguished vertices are equal.

Specializing to $\alpha=3$6 gives strengthened lower bounds for the classical first Zagreb index. In particular,

$\alpha=3$7

and choosing $\alpha=3$8, $\alpha=3$9 yields a strict improvement over the earlier inequality

$G=(V,E)$0

(Verma et al., 20 Aug 2025). Equality holds if and only if $G=(V,E)$1 is regular or belongs to specific degree-multiplicity classes such as $G=(V,E)$2, $G=(V,E)$3, $G=(V,E)$4, or related classes depending on the chosen specialization (Verma et al., 20 Aug 2025).

A complementary line of results derives sharp bounds from degree-class decompositions. Writing $G=(V,E)$5 for the number of vertices of degree $G=(V,E)$6 and

$G=(V,E)$7

one has, for $G=(V,E)$8 and $G=(V,E)$9,

$n=|V|$0

From the sign of the remainder terms, the following sharp bounds result (Vaidya et al., 2024):

• if $n=|V|$1 or $n=|V|$2, then

$n=|V|$3

with equality for any bi-degreed graph;

• if $n=|V|$4, then the same expression is a lower bound, again with equality for any bi-degreed graph.

The same paper proves refined bounds based on $n=|V|$5 and a modular decomposition

$n=|V|$6

leading to one-vertex-refinement extremals with $n=|V|$7 vertices of degree $n=|V|$8, one vertex of degree $n=|V|$9, and all remaining vertices of degree $M_1^{\alpha}(G)=\sum_{v\in V(G)} d(v)^{\alpha},$00 (Vaidya et al., 2024). In both the variance-enhanced and degree-class frameworks, increased degree heterogeneity contributes explicit correction terms, so the general index is controlled not only by the average degree but also by the spread and placement of the degrees (Verma et al., 20 Aug 2025).

4. Star sequences, generating functions, and recurrences

For integer indices, the first general Zagreb index admits a precise combinatorial representation through star counts. Let $M_1^{\alpha}(G)=\sum_{v\in V(G)} d(v)^{\alpha},$01 be the number of vertices of degree $M_1^{\alpha}(G)=\sum_{v\in V(G)} d(v)^{\alpha},$02, and let $M_1^{\alpha}(G)=\sum_{v\in V(G)} d(v)^{\alpha},$03 denote the number of centered $M_1^{\alpha}(G)=\sum_{v\in V(G)} d(v)^{\alpha},$04-stars:

$M_1^{\alpha}(G)=\sum_{v\in V(G)} d(v)^{\alpha},$05

Here $M_1^{\alpha}(G)=\sum_{v\in V(G)} d(v)^{\alpha},$06, and for $M_1^{\alpha}(G)=\sum_{v\in V(G)} d(v)^{\alpha},$07 the centered and uncentered counts coincide with the number of non-induced subgraphs isomorphic to $M_1^{\alpha}(G)=\sum_{v\in V(G)} d(v)^{\alpha},$08 (Bedratyuk et al., 2017).

The forward relation between degree frequencies and star counts is

$M_1^{\alpha}(G)=\sum_{v\in V(G)} d(v)^{\alpha},$09

and the inverse binomial transform is

$M_1^{\alpha}(G)=\sum_{v\in V(G)} d(v)^{\alpha},$10

Thus the frequency sequence and the star sequence are inverse to each other in a combinatorial sense (Bedratyuk et al., 2017).

The main representation for the general first Zagreb index uses the Stirling expansion

$M_1^{\alpha}(G)=\sum_{v\in V(G)} d(v)^{\alpha},$11

which yields

$M_1^{\alpha}(G)=\sum_{v\in V(G)} d(v)^{\alpha},$12

In the paper’s star notation this is

$M_1^{\alpha}(G)=\sum_{v\in V(G)} d(v)^{\alpha},$13

because $M_1^{\alpha}(G)=\sum_{v\in V(G)} d(v)^{\alpha},$14 and $M_1^{\alpha}(G)=\sum_{v\in V(G)} d(v)^{\alpha},$15 for $M_1^{\alpha}(G)=\sum_{v\in V(G)} d(v)^{\alpha},$16 (Bedratyuk et al., 2017). For $M_1^{\alpha}(G)=\sum_{v\in V(G)} d(v)^{\alpha},$17 one recovers

$M_1^{\alpha}(G)=\sum_{v\in V(G)} d(v)^{\alpha},$18

which matches the identity $M_1^{\alpha}(G)=\sum_{v\in V(G)} d(v)^{\alpha},$19.

These formulas lead to generating functions. The ordinary generating function

$M_1^{\alpha}(G)=\sum_{v\in V(G)} d(v)^{\alpha},$20

is rational with denominator

$M_1^{\alpha}(G)=\sum_{v\in V(G)} d(v)^{\alpha},$21

because

$M_1^{\alpha}(G)=\sum_{v\in V(G)} d(v)^{\alpha},$22

and $M_1^{\alpha}(G)=\sum_{v\in V(G)} d(v)^{\alpha},$23 for $M_1^{\alpha}(G)=\sum_{v\in V(G)} d(v)^{\alpha},$24 (Bedratyuk et al., 2017). The exponential generating function collapses to

$M_1^{\alpha}(G)=\sum_{v\in V(G)} d(v)^{\alpha},$25

Consequently, the sequence $M_1^{\alpha}(G)=\sum_{v\in V(G)} d(v)^{\alpha},$26 satisfies a linear recurrence of order $M_1^{\alpha}(G)=\sum_{v\in V(G)} d(v)^{\alpha},$27:

$M_1^{\alpha}(G)=\sum_{v\in V(G)} d(v)^{\alpha},$28

where $M_1^{\alpha}(G)=\sum_{v\in V(G)} d(v)^{\alpha},$29 are elementary symmetric sums (Bedratyuk et al., 2017). This recasts the index sequence as a finite-order linear dynamical object determined by the maximum degree.

5. Extremal graph classes and exact evaluations

For connected $M_1^{\alpha}(G)=\sum_{v\in V(G)} d(v)^{\alpha},$30-cyclic graphs, where the cyclomatic number is $M_1^{\alpha}(G)=\sum_{v\in V(G)} d(v)^{\alpha},$31, majorization provides a systematic extremal theory for $M_1^{\alpha}(G)=\sum_{v\in V(G)} d(v)^{\alpha},$32 on degree-sequence classes (Bianchi et al., 2013). Since $M_1^{\alpha}(G)=\sum_{v\in V(G)} d(v)^{\alpha},$33 is convex on $M_1^{\alpha}(G)=\sum_{v\in V(G)} d(v)^{\alpha},$34 for $M_1^{\alpha}(G)=\sum_{v\in V(G)} d(v)^{\alpha},$35 or $M_1^{\alpha}(G)=\sum_{v\in V(G)} d(v)^{\alpha},$36 and concave for $M_1^{\alpha}(G)=\sum_{v\in V(G)} d(v)^{\alpha},$37, the index is Schur-convex in the first regime and Schur-concave in the second (Bianchi et al., 2013). The paper determines explicit extremal degree sequences for connected $M_1^{\alpha}(G)=\sum_{v\in V(G)} d(v)^{\alpha},$38-cyclic graphs with $M_1^{\alpha}(G)=\sum_{v\in V(G)} d(v)^{\alpha},$39.

For trees ($M_1^{\alpha}(G)=\sum_{v\in V(G)} d(v)^{\alpha},$40), the maximal sequence is $M_1^{\alpha}(G)=\sum_{v\in V(G)} d(v)^{\alpha},$41, realized by the star, and the minimal integer sequence is $M_1^{\alpha}(G)=\sum_{v\in V(G)} d(v)^{\alpha},$42, realized by the path. Hence, for $M_1^{\alpha}(G)=\sum_{v\in V(G)} d(v)^{\alpha},$43 or $M_1^{\alpha}(G)=\sum_{v\in V(G)} d(v)^{\alpha},$44,

$M_1^{\alpha}(G)=\sum_{v\in V(G)} d(v)^{\alpha},$45

with equality at $M_1^{\alpha}(G)=\sum_{v\in V(G)} d(v)^{\alpha},$46 and $M_1^{\alpha}(G)=\sum_{v\in V(G)} d(v)^{\alpha},$47, respectively (Bianchi et al., 2013). For unicyclic graphs ($M_1^{\alpha}(G)=\sum_{v\in V(G)} d(v)^{\alpha},$48),

$M_1^{\alpha}(G)=\sum_{v\in V(G)} d(v)^{\alpha},$49

with equality at the cycle and at the “star-plus-edge” graph $M_1^{\alpha}(G)=\sum_{v\in V(G)} d(v)^{\alpha},$50 (Bianchi et al., 2013). For bicyclic graphs ($M_1^{\alpha}(G)=\sum_{v\in V(G)} d(v)^{\alpha},$51),

$M_1^{\alpha}(G)=\sum_{v\in V(G)} d(v)^{\alpha},$52

recovering known extremal bicyclic bounds when $M_1^{\alpha}(G)=\sum_{v\in V(G)} d(v)^{\alpha},$53 (Bianchi et al., 2013). When $M_1^{\alpha}(G)=\sum_{v\in V(G)} d(v)^{\alpha},$54, the lower-bound pattern is

$M_1^{\alpha}(G)=\sum_{v\in V(G)} d(v)^{\alpha},$55

while for $M_1^{\alpha}(G)=\sum_{v\in V(G)} d(v)^{\alpha},$56 the upper bound is obtained from one of several incomparable maximal sequences listed explicitly up to $M_1^{\alpha}(G)=\sum_{v\in V(G)} d(v)^{\alpha},$57 (Bianchi et al., 2013).

Exact formulas are also available for several standard graph families. For an $M_1^{\alpha}(G)=\sum_{v\in V(G)} d(v)^{\alpha},$58-regular graph,

$M_1^{\alpha}(G)=\sum_{v\in V(G)} d(v)^{\alpha},$59

and every bound discussed above becomes an equality (Verma et al., 20 Aug 2025). For the complete graph,

$M_1^{\alpha}(G)=\sum_{v\in V(G)} d(v)^{\alpha},$60

for the cycle,

$M_1^{\alpha}(G)=\sum_{v\in V(G)} d(v)^{\alpha},$61

for the path,

$M_1^{\alpha}(G)=\sum_{v\in V(G)} d(v)^{\alpha},$62

for the star,

$M_1^{\alpha}(G)=\sum_{v\in V(G)} d(v)^{\alpha},$63

and for the complete bipartite graph,

$M_1^{\alpha}(G)=\sum_{v\in V(G)} d(v)^{\alpha},$64

(Bedratyuk et al., 2017, Verma et al., 20 Aug 2025, Vaidya et al., 2024).

The same exact-evaluation program extends to lattice networks. For the hexagonal lattices of (Sarkar et al., 2019),

$M_1^{\alpha}(G)=\sum_{v\in V(G)} d(v)^{\alpha},$65

$M_1^{\alpha}(G)=\sum_{v\in V(G)} d(v)^{\alpha},$66

$M_1^{\alpha}(G)=\sum_{v\in V(G)} d(v)^{\alpha},$67

while for the triangular lattices,

$M_1^{\alpha}(G)=\sum_{v\in V(G)} d(v)^{\alpha},$68

$M_1^{\alpha}(G)=\sum_{v\in V(G)} d(v)^{\alpha},$69

$M_1^{\alpha}(G)=\sum_{v\in V(G)} d(v)^{\alpha},$70

Asymptotically, the index is dominated by interior degrees: degree $M_1^{\alpha}(G)=\sum_{v\in V(G)} d(v)^{\alpha},$71 for hexagonal lattices and degree $M_1^{\alpha}(G)=\sum_{v\in V(G)} d(v)^{\alpha},$72 for triangular lattices, with boundary terms of lower order (Sarkar et al., 2019).

6. Related indices, coindices, and applications

The first general Zagreb index is tightly linked to several other degree-based invariants. In the coindex direction, the identity

$M_1^{\alpha}(G)=\sum_{v\in V(G)} d(v)^{\alpha},$73

holds for every real $M_1^{\alpha}(G)=\sum_{v\in V(G)} d(v)^{\alpha},$74 (Verma et al., 20 Aug 2025), while in the $M_1^{\alpha}(G)=\sum_{v\in V(G)} d(v)^{\alpha},$75 notation one has, for integer $M_1^{\alpha}(G)=\sum_{v\in V(G)} d(v)^{\alpha},$76,

$M_1^{\alpha}(G)=\sum_{v\in V(G)} d(v)^{\alpha},$77

(Vaidya et al., 2024). These formulas let bounds for the first general Zagreb index propagate directly to generalized coindices.

The sharpened lower bounds for $M_1^{\alpha}(G)=\sum_{v\in V(G)} d(v)^{\alpha},$78 also imply stronger lower bounds for the second Zagreb index through the inequality

$M_1^{\alpha}(G)=\sum_{v\in V(G)} d(v)^{\alpha},$79

and they sharpen the spectral-radius estimate

$M_1^{\alpha}(G)=\sum_{v\in V(G)} d(v)^{\alpha},$80

by improving the lower bound on $M_1^{\alpha}(G)=\sum_{v\in V(G)} d(v)^{\alpha},$81 (Verma et al., 20 Aug 2025). The same paper further states Nordhaus–Gaddum-type identities for $M_1^{\alpha}(G)=\sum_{v\in V(G)} d(v)^{\alpha},$82, $M_1^{\alpha}(G)=\sum_{v\in V(G)} d(v)^{\alpha},$83, and $M_1^{\alpha}(G)=\sum_{v\in V(G)} d(v)^{\alpha},$84 which immediately convert improved $M_1^{\alpha}(G)=\sum_{v\in V(G)} d(v)^{\alpha},$85 bounds into improved inequalities for these indices and their coindices (Verma et al., 20 Aug 2025).

A different connection appears in graph transformations. For the subdivision graph $M_1^{\alpha}(G)=\sum_{v\in V(G)} d(v)^{\alpha},$86,

$M_1^{\alpha}(G)=\sum_{v\in V(G)} d(v)^{\alpha},$87

and therefore

$M_1^{\alpha}(G)=\sum_{v\in V(G)} d(v)^{\alpha},$88

with equality if and only if $M_1^{\alpha}(G)=\sum_{v\in V(G)} d(v)^{\alpha},$89 is regular (Ilić et al., 2011). This reinforces the recurring fact that regular graphs are the exact extremals for the basic normalized inequalities involving first Zagreb-type quantities.

The literature also develops adjacent degree-based generalizations. For triangle- and quadrangle-free graphs, the generalized first leap Zagreb index is defined by

$M_1^{\alpha}(G)=\sum_{v\in V(G)} d(v)^{\alpha},$90

where $M_1^{\alpha}(G)=\sum_{v\in V(G)} d(v)^{\alpha},$91 is the $M_1^{\alpha}(G)=\sum_{v\in V(G)} d(v)^{\alpha},$92-distance degree, and the identity

$M_1^{\alpha}(G)=\sum_{v\in V(G)} d(v)^{\alpha},$93

holds under the same structural assumptions (Vaidya et al., 2024). The leap-index theory parallels the first general Zagreb theory: it has degree-class decompositions, sharp convexity/concavity bounds, modular refinements, and explicit extremal configurations (Vaidya et al., 2024).

Chemical applications are represented in the regression study on benzenoid hydrocarbons. For entropy $M_1^{\alpha}(G)=\sum_{v\in V(G)} d(v)^{\alpha},$94 of $M_1^{\alpha}(G)=\sum_{v\in V(G)} d(v)^{\alpha},$95 benzenoid hydrocarbons, the paper reports

$M_1^{\alpha}(G)=\sum_{v\in V(G)} d(v)^{\alpha},$96

while the regression based on $M_1^{\alpha}(G)=\sum_{v\in V(G)} d(v)^{\alpha},$97 is

$M_1^{\alpha}(G)=\sum_{v\in V(G)} d(v)^{\alpha},$98

For boiling point $M_1^{\alpha}(G)=\sum_{v\in V(G)} d(v)^{\alpha},$99 of ${}^\lambda M_1=\sum_{i=1}^n d_i^{\,2\lambda}$00 benzenoid hydrocarbons, the reported models are

${}^\lambda M_1=\sum_{i=1}^n d_i^{\,2\lambda}$01

and

${}^\lambda M_1=\sum_{i=1}^n d_i^{\,2\lambda}$02

In both cases, the classical first Zagreb index outperforms the first leap Zagreb index in correlation (Vaidya et al., 2024).

Taken together, these results place the first general Zagreb index at the center of a broad degree-based framework. Its basic form is a power sum of vertex degrees, but its theory now includes sharp average-degree and maximum-degree bounds, variance-corrected inequalities, explicit equality classes, combinatorial inversion through star counts, recurrence theory, extremal descriptions for ${}^\lambda M_1=\sum_{i=1}^n d_i^{\,2\lambda}$03-cyclic graphs, exact formulas on lattice families, and systematic transfers to coindices, spectral parameters, and chemically motivated regression models (Ilić et al., 2011, Bedratyuk et al., 2017, Bianchi et al., 2013, Sarkar et al., 2019, Vaidya et al., 2024, Verma et al., 20 Aug 2025).