Euler Sombor index for trees with given diameter

SUMMARY OF THE EULER–SOMBOR INDEX PROJECT

1. Research Problem

The project studies the Euler–Sombor index of finite trees with prescribed order $n$ and diameter $d$.

For a tree $T$, the Euler–Sombor index is

$$\operatorname{EUS}(T) = \sum_{uv\in E(T)} \sqrt{d(u)^2+d(v)^2+d(u)d(v)}.$$

The principal objective is to determine the largest, second largest, third largest, and fourth largest Euler–Sombor values among all $n$-vertex trees having diameter $d$, together with the corresponding extremal tree structures.

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2. Main Notation

Set

$$q=n-d$$

and define

$$\Phi(a,b)=\sqrt{a^2+b^2+ab}.$$

This notation makes the extremal values considerably easier to express.

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3. First Maximum

For $d\ge4$, the known maximum is attained by the broom-type tree $T_1$, whose diametral path has degree sequence

$$1,q+1,2,\ldots,2,1.$$

All surplus leaves are concentrated at one vertex adjacent to an endpoint of the diametral path.

Its Euler–Sombor index is

$$M_1 = q\Phi(q+1,1) +\Phi(q+1,2) +(d-3)\Phi(2,2) +\Phi(2,1),$$

or explicitly,

$$\boxed{ M_1= q\sqrt{q^2+3q+3} +\sqrt{q^2+4q+7} +2(d-3)\sqrt3+\sqrt7 }.$$

This part is supported by the established fixed-diameter extremal theory.

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4. Second Maximum

The second-largest value is attained by $T_2$, whose diametral path has degree pattern

$$1,2,q+1,2,\ldots,2,1.$$

Its value is

$$M_2 = (q-1)\Phi(q+1,1) +2\Phi(q+1,2) +(d-4)\Phi(2,2) +2\Phi(2,1),$$

hence

$$\boxed{ M_2= (q-1)\sqrt{q^2+3q+3} +2\sqrt{q^2+4q+7} +2(d-4)\sqrt3+2\sqrt7 }.$$

The strict inequality

$$M_1>M_2$$

follows from

$$M_1-M_2 = \sqrt{q^2+3q+3} -\sqrt{q^2+4q+7} +2\sqrt3-\sqrt7>0.$$

Thus the first two levels are already firmly established.

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5. Proposed Third Maximum

The next extremal configuration is the two-branching-vertex structure $T_3$, with diametral-path pattern

$$1,q,2,\ldots,2,3,1.$$

The degree-$q$ vertex carries $q-2$ additional leaves, while the degree-$3$ vertex carries one additional leaf.

The relevant edge multiplicities are

$$m_{1,q}=q-1,\qquad m_{2,q}=1,\qquad m_{2,2}=d-4,$$ $$m_{2,3}=1,\qquad m_{1,3}=2.$$

Consequently,

$$M_3 = (q-1)\Phi(q,1) +\Phi(q,2) +(d-4)\Phi(2,2) +\Phi(2,3) +2\Phi(1,3),$$

or

$$\boxed{ M_3= (q-1)\sqrt{q^2+q+1} +\sqrt{q^2+2q+4} +2(d-4)\sqrt3 +\sqrt{19} +2\sqrt{13} }.$$

The analysis gives

$$M_2>M_3.$$

Therefore the proposed third level is below the known second maximum.

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6. Proposed Fourth Maximum

For $d\ge5$, the proposed fourth extremal tree $T_4$ has path pattern

$$1,2,3,2,\ldots,2,q,1.$$

Again, the degree-$q$ vertex carries $q-2$ extra leaves and the degree-$3$ vertex carries one extra leaf.

Its value is

$$M_4 = (q-1)\Phi(q,1) +\Phi(q,2) +(d-5)\Phi(2,2) +2\Phi(2,3) +\Phi(1,3) +\Phi(1,2),$$

namely

$$\boxed{ M_4= (q-1)\sqrt{q^2+q+1} +\sqrt{q^2+2q+4} +2(d-5)\sqrt3 +2\sqrt{19} +\sqrt{13} +\sqrt7 }.$$

Importantly,

$$M_3-M_4 = 2\sqrt3+\sqrt{13}-\sqrt{19}-\sqrt7 >0.$$

Numerically,

$$M_3-M_4\approx0.0650026366.$$

Hence

$$\boxed{M_3>M_4}.$$

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7. Special Case $d=4$

For diameter $4$, the fourth candidate requires a slightly different configuration.

The proposed pattern is

$$1,q,3,2,1$$

up to reversal.

Its value is

$$\boxed{ M_4^{(d=4)} = (q-1)\Phi(q,1) +\Phi(q,3) +\Phi(3,2) +\Phi(2,1) +\Phi(3,1) }.$$

Equivalently,

$$M_4^{(d=4)} = (q-1)\sqrt{q^2+q+1} +\sqrt{q^2+3q+9} +\sqrt{19} +\sqrt7 +\sqrt{13}.$$

The candidate ordering remains

$$M_1>M_2>M_3>M_4^{(d=4)},$$

although the last inequality requires a particularly careful radical comparison in a formal proof.

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8. Diameter $d=3$

Diameter $3$ is structurally different.

Every tree of diameter $3$ is a double star

$$S_{a,n-a}, \qquad 2\le a\le\left\lfloor\frac n2\right\rfloor.$$

Its Euler–Sombor index is

$$EUS(S_{a,n-a}) = (a-1)\Phi(a,1) +(n-a-1)\Phi(n-a,1) +\Phi(a,n-a).$$

The known ordering is

$$EUS(S_{2,n-2}) > EUS(S_{3,n-3}) > EUS(S_{4,n-4}) >\cdots.$$

Therefore:

• the maximum corresponds to $a=2$;
• the second maximum corresponds to $a=3$;
• the third maximum corresponds to $a=4$;
• the fourth maximum corresponds to $a=5$.

These exist when respectively

$$n\ge4,\quad n\ge6,\quad n\ge8,\quad n\ge10.$$

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9. Diameter $d=2$

For $d=2$, the only tree is the star $K_{1,n-1}$.

Therefore there are no distinct second, third, or fourth largest values.

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10. Structural Principle Behind the Extremal Trees

The central structural idea is that Euler–Sombor maximization favors concentration of degree.

For a tree,

$$\sum_{v\in V(T)}(d(v)-2)=-2.$$

Equivalently, if $L(T)$ denotes the number of leaves,

$$\sum_{v\in B(T)}(d(v)-2)=L(T)-2,$$

where $B(T)$ is the set of branching vertices.

This identity provides the degree-excess budget.

The extremal analysis therefore proceeds by:

1. Fixing a diametral path.
2. Moving non-path branches toward suitable path vertices.
3. Concentrating pendant leaves.
4. Comparing the resulting degree distributions.
5. Using monotonicity of the Euler–Sombor edge weight.
6. Reducing the problem to a finite collection of degree/path configurations.

The basic incremental quantity is

$$\Psi(x,y) = \Phi(x,y)-\Phi(x-1,y) = \frac{2x+y-1} {\Phi(x,y)+\Phi(x-1,y)}.$$

Its monotonic behavior provides the mechanism for proving that transferring degree toward a larger branching vertex increases the index.

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11. Candidate Hierarchy

For $d\ge5$, the proposed hierarchy is

$$\boxed{ M_1>M_2>M_3>M_4 }.$$

The corresponding structures are

$$T_1: \quad 1,q+1,2,\ldots,2,1,$$ $$T_2: \quad 1,2,q+1,2,\ldots,2,1,$$ $$T_3: \quad 1,q,2,\ldots,2,3,1,$$ $$T_4: \quad 1,2,3,2,\ldots,2,q,1.$$

Thus the hierarchy reflects a transition from:

one highly concentrated branching vertex

to

two branching vertices with increasingly less favorable positioning.

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12. Computational Verification

Exhaustive enumeration of non-isomorphic trees using computational graph enumeration was carried out for several cases.

Representative results were:

$n$	$d$	1st	2nd	3rd	4th
7	4	25.149083	25.039659	23.140003	23.017457
8	4	34.625908	34.484792	30.609231	30.440691
10	5	49.572250	49.407563	43.550159	43.485156
12	6	66.522416	66.339584	58.496500	58.431497
15	8	88.939437	88.742234	78.910767	78.845765
15	12	52.861896	52.752472	50.852816	50.787813

The proposed formulas agree with the computational rankings in all tested cases.

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13. The Most Important Mathematical Issue Remaining

The first and second maxima have established theoretical support.

The proposed $T_3$ and $T_4$ configurations are strongly supported by:

• structural degree-excess arguments;
• transfer inequalities;
• exact symbolic comparisons;
• exhaustive computational tests;
• consistency with the known extremal framework.

However, the publication-critical step is the complete global exclusion theorem:

$$\boxed{ T\not\cong T_1,T_2,T_3,T_4 \quad\Longrightarrow\quad EUS(T)<M_4. }$$

This must be proved against every other admissible tree structure, not merely against the most obvious competitors.

In particular, one must rigorously exclude:

• trees with three or more branching vertices;
• alternative degree partitions;
• different positions of the branching vertices;
• non-pendant off-path branches;
• boundary configurations for small $n,d$;
• exceptional cases where $q$ is small.

Therefore the present work should be regarded as a nearly complete extremal framework, but the universal fourth-level exclusion should not yet be stated as a fully established theorem without that final proof.

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14. Overall Research Conclusion

The project has established a coherent extremal picture for the Euler–Sombor index of trees with prescribed diameter.

The central phenomenon is:

$$\boxed{\text{Euler–Sombor maximization strongly favors concentrated branching.}}$$

The first maximum is obtained by concentrating essentially all available degree excess at one branching vertex. The second maximum is obtained by a small displacement of that concentration. At the next levels, the extremal structures appear to split the branching into a dominant degree-$q$ vertex and a secondary degree-$3$ vertex, with the precise location of the latter determining the third and fourth levels.

Thus the project has reached the stage where the principal remaining task is not discovering plausible candidates, but proving the global exclusion and uniqueness of all competitors below the fourth level.

Once that exclusion lemma is established, the work can be upgraded from a computationally and structurally supported manuscript into a complete theorem on the first four maximum Euler–Sombor values for trees with prescribed diameter.