Disproof of a conjecture on the minimum Wiener index of signed trees

The Wiener index of a connected graph is the sum of distances between all unordered pairs of vertices. Sam Spiro [The Wiener index of signed graphs, Appl. Math. Comput., 416(2022)126755] recently introduced the Wiener index for a signed graph and conjectured that the path $P_n$ with alternating signs has the minimum Wiener index among all signed trees with $n$ vertices. By constructing an infinite family of counterexamples, we prove that the conjecture is false whenever $n$ is at least 30.Comment: 7 pages, 2 figure

Certain topological indices and related polynomials for polysaccharides

A polysaccharide is a large molecule made of many smaller monosaccharides. Monosaccharides are simple sugars, like glucose. Special enzymes bind these small monomers together creating large sugar polymers or polysaccharides. A polysaccharide is also called a glycan. Starch, glycogen, and cellulose are examples of polysaccharides. Depending on their structure, polysaccharides can have a wide variety of functions in nature. Some polysaccharides are used for storing energy, some for sending cellular messages, and others for providing support to cells and tissues. In the present work, we focus on the polysaccharides, namely, amylose and blue starch-iodine complex. Several topological indices and polynomials are determined in view of edge dividing methods. Also, depict their graphic behavior.Publisher's Versio

Amplified eccentric connectivity index of graphs

A new distance based graphical index, coined as amplified eccentric connectivity index, has been established and the formulae to calculate the amplified eccentric connectivity index of some standard graphs, Dutch windmill graph and molecular graph of cycloalkenes has been computed. Also, in the case of boiling points of primary and secondary amines, the study shows that the amplified eccentric connectivity index gives a greater correlation of 98%, when compared to the Wiener and Eccentric connectivity indices

Mean Sombor index

A Special Volume on Chemical Graph Theory in Memory of Nenad TrinajsticWe introduce a degree–based variable topological index inspired on the power (or generalized) mean. We name this new index as the mean Sombor index: SOα(G) = P uv∈E(G) [(d α u + d α v ) /2]1/α. Here, uv denotes the edge of the graph G connecting the vertices u and v, du is the degree of the vertex u, and α ∈ R\{0}. We also consider the limit cases mSOα→0(G) and SOα→±∞(G). Indeed, for given values of α, the mean Sombor index is related to well-known opological indices such as the inverse sum indeg index, the reciprocal Randic index, the first Zagreb index, the Stolarsky–Puebla index and several ´Sombor indices. Moreover, through a quantitative structure property relationship (QSPR) analysis we show that mSOα(G) correlates well with several physicochemical properties of octane isomers. Some mathematical properties of the mean Sombor index as well as bounds and new relationships with known topological indices are also discussed.J.A.M.-B. acknowledges financial support from CONACyT (Grant No. A1-S-22706) and BUAP (Grant No. 100405811VIEP2021) .E.D.M. and J.M.R. were supported by a grant from Agencia Estatal de Investigación (PID 2019-106433GBI00 / AEI / 10.13039 / 501100011033), Spain. J.M.R. was supported by the Madrid Government (Comunidad de Madrid-Spain) under the Multiannual Agreement with UC3M in the line of Excellence of University Professors (EPUC3M23), and in the context of the VPRICIT (Regional Programme of Research and Technological Innovation)