On the spectral radius of VDB graph matrices

: Introduction/purpose: Vertex-degree-based (VDB) graph matrices form a special class of matrices, corresponding to the currently much investigated vertex-degree-based (VDB) graph invariants. Some spectral properties of these matrices are investigated. Results: Generally valid sharp lower and upper bounds are established for the spectral radius of any VDB matrix. The equality cases are characterized. Several earlier published results are shown to be special cases of the presently reported bounds. Conclusion: The results of the paper contribute to the general spectral theory of VDB matrices, as well as to the general theory of VDB graph invariants.

In the present-day mathematical and chemical literature, a large number, well over hundred, of degree-based graph invariants of the form ( ) are being studied, where ( , ) f x y is an appropriately chosen function with the property In chemistry, molecular physics, pharmacology, and elsewhere, these graph invariants found a great variety of applications, and are usually referred to as "topological indices" or "molecular structuredescriptors" (Gutman, 2013;Todeschini & Consonni, 2009;Kulli, 2020).Instead of "vertex-degree-based" the abbreviation VDB is often used (Rada, 2014;Li et al, 2021;Monsalve & Rada, 2022).
Let the vertices of the graph G be labelled as  ( ; ) M f G .Further on, they will be denoted by 1 2 , , , In order to prove our main result, Theorem 1, we need an auxiliary lemma.
Lemma 1.Let Proof.By definition of the matrix ( ; ) M f G , its diagonal elements are always equal to zero.From this, Eq. ( 2) follows straightforwardly.
In order to arrive at Eq. ( 3), note that the sum of k-th powers of the eigenvalues is equal to the trace (sum of diagonal elements) of the k-th power of the respective matrix.Thus, ( ) where we used the above specified definition of the elements of the VDB matrix Note that the above lemma is a direct generalization of Lemma 1 in (Gutman, 2021), stated for a special case of the function f in Eq. ( 1), namely for We are now prepared to state our main result Theorem 1.Let G be a connected graph of the order n, and let 1  be the spectral radius of its VDB matrix The equality on the left-hand side holds if and only if G is regular.The equality on the right-hand side holds if n GK  .

Proof of Theorem 1
Lower bound.We proceed in an analogous manner as in the proof of Lemma 2 in (Gutman, 2021).Thus, in view of the Rayleigh-Ritz variational principle, for an n-dimensional column-vector with equality if and only if (1,1,...,1) T = is an eigenvector of ( ; ) M f G , corresponding to the eiigenvalue 1  .As it is well known (Brualdi & Cvetković, 2008;Cvetković et al, 2010), this happens if and only if the graph G is regular.
The lower bound for the spectral radius follows directly from Eq. ( 5).
Upper bound.Eq. ( 2) can be rewritten as Using the Cauchy-Schwarz inequality, we get The upper bound for the spectral radius is followed by Eq. ( 3).The equality in (6) happens if and only if 2 ( 1, 1) EDITORIAL NOTE: The author of this article, Ivan Gutman, is a current member of the Editorial Board of the Military Technical Courier.Therefore, the Editorial Team has ensured that the double blind reviewing process was even more transparent and more rigorous.The Team made additional effort to maintain the integrity of the review and to minimize any bias by having another associate editor handle the review procedure independently of the editorauthor in a completely transparent process.The Editorial This paper concerns simple connected graphs.Let G be such a graph.Its vertex and edges sets are () VG and () EG , respectively, whereas its order (number of vertices) and size (number of edges) are the edge of G connecting the vertices u and v.