DISTANCE $\tau$-POLYNOMIAL AND EDGE DISTANCE $\tau$-POLYNOMIAL OF GRAPHS

Veena Mathad; Pavithra M

DISTANCE $\tau$-POLYNOMIAL AND EDGE DISTANCE $\tau$-POLYNOMIAL OF GRAPHS

Abstract

The distance $\tau$-polynomial $\mathbb{D}(G, x)$ of a graph $G=(V, E)$ is defined as $\mathbb{D}(G, x)=\displaystyle\sum_{k=0}^{\binom{n}{2}}\tau_kx^k$, where $\tau_k$ is the number of vertices with $\tau_G(u)=\displaystyle \sum_{x \in V(G)}{d_G(u,x)}= k, \forall u \in V(G)$. In this paper, we seek to find the distance $\tau$-polynomial and similarly, edge distance $\tau$-polynomial of some graph families and graph operations.

Keywords

References

[1] Albina, A. and Mary, U., (2018), A study on dominator coloring of friendship and barbell graphs, International Journal of Mathematics And its Applications, 6(4), pp. 99-105.
[2] Akbari, S., Alikhani, S. and Peng, Y. H., (2010), Characterization of graphs using domination polynomials, European journal of combinatorics, 31(7), pp. 1714-1724.
[3] Alfeche, F. L., Barraza, V. and Canoy, S., (2023), Closeness centrality of vertices in graphs under some operations, European Journal of Pure and Applied Mathematics, 16(3), pp. 1406-1420.
[4] Arundhadhi, R. and Ilayarani, V., (2017), Total coloring of closed helm, flower and bistar graph family, International Journal of Scientific and Research Publications, 7(7), pp. 616-621.
[5] Balakrishnan, R. and Ranganathan, K., (2012), A textbook of graph theory, Springer Science and Business Media.
[6] Basavanagoud, B. and Sayyed, M., (2021), Hub Polynomial of a Graphs, International Journal of Applied Engineering Research, 16(3), pp. 166-173.
[7] Das, K. C. and Liu, M., (2016), Complete split graph determined by its (signless) Laplacian spectrum, Discrete Applied Mathematics, 205, pp. 45-51.
[8] Eballe, R. G., Balingit, C. M. R., Cabahug Jr, I. S., Flores, A. L. V., Lumpayao, S. M. B., Pe˜nalosa, B. D., Tampipi, G. A. L. and Villarta, C. A., (2023), Closeness centrality in graph products, Advances and Applications in Discrete Mathematics, 39(1), pp. 29-41.

[9] Eballe, R. G., and Cabahug, I., (2021), Closeness centrality of some graph families, International Journal of Contemporary Mathematical Sciences, 16(4), pp. 127-134.
[10] Farell, E. J., (1997), A note on the clique polynomial and its relation to other graph polynomials, J. Math. Sci. Calcutta, 8, pp. 97-102.
[11] Fernau, H., Ryan, J. F. and Sugeng, K. A., (2008), A sum labelling for the generalised friendship graph, Discrete Mathematics, 308(5-6), pp. 734-740.
[12] Harary, F., (1969), Graph Theory, Addison Wesley, Reading Mass.
[13] Hosoya, H., (1988), On some counting polynomials in chemistry, Discrete applied mathematics, 19(1-3), pp. 239-257.
[14] Kulli, V. R., (2012), College Graph Theory, Vishwa Int. Publ., Gulbarga, India, 1.
[15] Maulidia, A. R., (2021), May. Elegant labeling of sun graphs and helm graphs, In Journal of Physics: Conference Series (Vol. 1872, No. 1, p. 012008). IOP Publishing.
[16] Mathad, V. and Pavithra, M., (2025), Closeness Centrality Weight and Edge Closeness Centrality Weight of Graphs, Dynamics of Continuous, Discrete and Impulsive Systems - Series B: Applications and Algorithms, 32, pp.109-124. [17] Mathad, V. and Pavithra, M., (2025), Closeness Centrality Weight of Graphs Under some Graph Operations, Communications in Mathematics and Applications, 16(2).
[18] Meena, S. and Vaithilingam, K., (2012), Prime labeling for some fan related graphs, International Journal of Engineering Research and Technology (IJERT), 9, pp. 2278-0181.
[19] Noy, M., (2003), Graphs determined by polynomial invariants, Theoretical Computer Science, 307(2), pp. 365-384.
[20] Padmapriya, P. and Mathad, V., (2017), The eccentric-distance sum of some graphs, Electronic Journal of Graph Theory and Applications (EJGTA), 5(1), pp. 51-62.
[21] Padmapriya, P. and Mathad, V., (2020), Eccentricity based topological indices of some graphs, TWMS Journal of Applied and Engineering Mathematics, 10(4), pp. 1084-1095.
[22] Prajapati, U. M. and Raval, K. K., (2016), Product Cordial Graph in the Context of Some Graph Operations on Gear Graph, Open Journal of Discrete Mathematics, 6(4), pp. 259-267.

Details

Primary Language

English

Subjects

Combinatorics and Discrete Mathematics (Excl. Physical Combinatorics)

Journal Section

Research Article

Authors

Veena Mathad
0000-0002-6621-9596
India

Pavithra M ^*
0009-0001-9689-9115
India

Publication Date

July 4, 2026

Submission Date

March 23, 2025

Acceptance Date

October 3, 2025

Published in Issue

Year 2026 Volume: 16 Number: 7

IZ

https://izlik.org/JA57DY54DA

Cite

RIS / Bibtex

APA

Mathad, V., & M, P. (2026). DISTANCE $\tau$-POLYNOMIAL AND EDGE DISTANCE $\tau$-POLYNOMIAL OF GRAPHS. TWMS Journal of Applied and Engineering Mathematics, 16(7), 853-864. https://izlik.org/JA57DY54DA

AMA

1.Mathad V, M P. DISTANCE $\tau$-POLYNOMIAL AND EDGE DISTANCE $\tau$-POLYNOMIAL OF GRAPHS. JAEM. 2026;16(7):853-864. https://izlik.org/JA57DY54DA

Chicago

Mathad, Veena, and Pavithra M. 2026. “DISTANCE $\tau$-POLYNOMIAL AND EDGE DISTANCE $\tau$-POLYNOMIAL OF GRAPHS”. TWMS Journal of Applied and Engineering Mathematics 16 (7): 853-64. https://izlik.org/JA57DY54DA.

EndNote

Mathad V, M P (July 1, 2026) DISTANCE $\tau$-POLYNOMIAL AND EDGE DISTANCE $\tau$-POLYNOMIAL OF GRAPHS. TWMS Journal of Applied and Engineering Mathematics 16 7 853–864.

IEEE

[1]V. Mathad and P. M, “DISTANCE $\tau$-POLYNOMIAL AND EDGE DISTANCE $\tau$-POLYNOMIAL OF GRAPHS”, JAEM, vol. 16, no. 7, pp. 853–864, July 2026, [Online]. Available: https://izlik.org/JA57DY54DA

ISNAD

Mathad, Veena - M, Pavithra. “DISTANCE $\tau$-POLYNOMIAL AND EDGE DISTANCE $\tau$-POLYNOMIAL OF GRAPHS”. TWMS Journal of Applied and Engineering Mathematics 16/7 (July 1, 2026): 853-864. https://izlik.org/JA57DY54DA.

JAMA

1.Mathad V, M P. DISTANCE $\tau$-POLYNOMIAL AND EDGE DISTANCE $\tau$-POLYNOMIAL OF GRAPHS. JAEM. 2026;16:853–864.

MLA

Mathad, Veena, and Pavithra M. “DISTANCE $\tau$-POLYNOMIAL AND EDGE DISTANCE $\tau$-POLYNOMIAL OF GRAPHS”. TWMS Journal of Applied and Engineering Mathematics, vol. 16, no. 7, July 2026, pp. 853-64, https://izlik.org/JA57DY54DA.

Vancouver

1.Veena Mathad, Pavithra M. DISTANCE $\tau$-POLYNOMIAL AND EDGE DISTANCE $\tau$-POLYNOMIAL OF GRAPHS. JAEM [Internet]. 2026 Jul. 1;16(7):853-64. Available from: https://izlik.org/JA57DY54DA