In this paper, we examine a class of vertex-degree-based (VDB) graph invariants. Let G be a simple graph with n vertices and m edges. Let V(G) and E(G) be its vertex and edge sets, respectively. Then |V(G)|=n and |E(G)|=m. The edge of the graph G, connecting the vertices u and v, will be denoted by uv. The degree d_u of the vertex u is the number of its first neighbors.

The graph in which any two vertices are adjacent is said to be complete and is denoted by K_n. It has m=n(n-1)/2 edges. Its complement, denoted by , is the edgeless graph, with m=0.

For additional details of graph theory, see (Harary, 1969; Bondy & Murty, 1976).

In the recent mathematical and chemical literature, a large number of graph invariants of the form

(1)

are studied, where f is a pertinently chosen function with the property f(x,y)=f(y,x); for details, see (Gutman, 2023) and the references cited therein. The quantities defined via Eq. (1) are usually referred to as vertexdegree-based (VDB) graph invariants. Of these, one of the oldest is the first Zagreb index (Gutman & Trinajstić, 1972; Gutman & Das, 2004):

whereas one of the most recent ones is the Sombor index (Gutman, 2021; Liu et al, 2022):

According to Fajtlowicz (Fajtlowicz, 1988), the temperature of the vertex u of a graph with n vertices is defined as

(2)

where one should recall that in the case of n-vertex graphs,

Directly from this definition, it follows that

The equality on the left-hand side holds if , whereas the righthand side equality holds if either or .

In Eq. (1), by replacing the vertex degrees with vertex temperatures, one obtains the respective temperature VDB graph invatiants, namely:

Such are the temperature first Zagreb index

(3)

and the temperature Sombor index

Several other temperature VDB graph invariants were studied in the literature (Narayankar et al, 2018; Kahsay et al, 2018; Kulli, 2019a; Kulli, 2019b; Kulli, 2021).

The temperature Sombor index was first considered by Kulli (Kulli, 2022). In this paper, we establish a few more of its properties.

Bearing in mind that for all vertices of any n-vertex graph, , directly from Eq. (2), we obtain:

Substitutingthis into Eq. (3) yields

(4)

where we used the identity (Gutman, 2015)

which holds for any quantity g determined by the vertex u. Note that TM₁ had to be be divided by n³ because the maximum possible value of .

In connection with formula (4), one should note that for k=1 and k=2, the term is equal to the well-known and much studied VDB invariants – the first Zagreb index M₁ and the so-called forgotten index F (Furtula & Gutman, 2015), respectively. The same term for k=3 and k=4 coincides with the VBD invariants Y and S, recently introduced in (Alameri et al, 2020) and (Nagarajan et al, 2021), respectively.

Therefore, , which is an approximation that would satisfy all practical applications of the temperature first Zagreb index. A somewhat better, yet more perplexed approximation would be .

It is evident from Eq. (2) that the temperature of a vertex is a monotonously increasing function of the respective vertex degree. Therefore, by deleting an adge from the graph G, some of its vertex temperatures must decrease, and no vertex temperature will increase. This implies,

(5)

From relation (5), we immediately conclude the following:

(1) The complete graph and its complement have the maximum and minimum temperature Sombor indices, i.e.,

(2) The connected graph with the minimum value of TSO must be a tree.

(3) Based on a general result for VDB graph invariants (Cruz & Rada, 2019), the trees with the maximum and minimum temperature Sombor indices are the star and the path, respectively.

In what follows, we use the well-known inequality

valid for a,b ≥ 0, with the left-hand side equality if a=b, and the right-hand side inequality in the irrelevant case a=b=0. Applying it to TSO, we get

i.e.,

With the left-hand side equality if and only if the graph G is regular, i.e., if all its vertices have mutually equal degrees.

Bearing in mind Eq. (4), we get

(6)

From (6), we immediately obtain the following lower bounds for TSO.

(7)

or, better, but more complicated,

(8)

The equality in (7) and (8) holds if .

In order to get an upper bound for TSO, we modify the right-hand side of (6) as

from which it follows