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All Journal BIMASTER CAUCHY: Jurnal Matematika Murni dan Aplikasi Journal of the Indonesian Mathematical Society BAREKENG: Jurnal Ilmu Matematika dan Terapan Zero : Jurnal Sains, Matematika, dan Terapan International Journal of Computing Science and Applied Mathematics-IJCSAM
Raventino Raventino
Universitas Tanjungpura
Agriculture, Biological Sciences & Forestry Chemistry Computer Science & IT Control & Systems Engineering Decision Sciences, Operations Research & Management Economics, Econometrics & Finance Energy Engineering Mathematics Mechanical Engineering Physics Transportation

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BILANGAN DIACHROMATIC PADA GRAF BINTANG Raventino Raventino; Nilamsari Kusumastuti; Fransiskus Fran
Bimaster : Buletin Ilmiah Matematika, Statistika dan Terapannya Vol 10, No 3 (2021): Bimaster : Buletin Ilmiah Matematika, Statistika dan Terapannya
Publisher : FMIPA Universitas Tanjungpura

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.26418/bbimst.v10i3.47401

Abstract

Pewarnaan lengkap pada suatu graf G adalah pewarnaan titik dengan syarat setiap pasangan warna muncul minimal satu kali pada G. Maksimum banyaknya warna yang digunakan pada pewarnaan lengkap suatu graf tidak berarah G yang dinotasikan dengan ψ(G) disebut bilangan achromatic. Pada penelitian ini dibahas perluasan dari bilangan achromatic yaitu bilangan diachromatic, khususnya bilangan diachromatic graf bintang berarah K1,n. Graf bintang K1,n adalah graf yang memuat satu titik pusat yang berderajat n dan bertetangga dengan n daun. Bilangan diachromatic yang dinotasikan dengan dac(G), adalah maksimum banyaknya warna yang digunakan pada pewarnaan lengkap suatu graf berarah G. Pada penelitian ini diperoleh bahwa banyaknya warna (dinotasikan w) yang dapat digunakan dalam pewarnaan lengkap graf berarah G adalah bilangan bulat yang memenuhi permutasi  dari w (P2w) yang tidak lebih dari atau sama dengan banyaknya sisi di graf G. Selain itu didapat bahwa bilangan diachromatic pada graf bintang berarah K1,n  adalah dac(K1,n ) = 2.Kata Kunci: pewarnaan titik, pewarnaan lengkap, maksimum banyaknya warna.
Diachromatic Number of Some Acyclic Digraphs Raventino, Raventino; Susanti, Yeni
Journal of the Indonesian Mathematical Society Vol. 31 No. 3 (2025): SEPTEMBER
Publisher : IndoMS

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.22342/jims.v31i3.1710

Abstract

A vertex coloring that ensures every pair of different colors is represented at least once is termed complete coloring. The diachromatic number of an acyclic digraph denotes the maximum number of colors required for its complete coloring. This study delves into the diachromatic numbers of lobster digraphs, fireworks digraphs, banana tree digraphs, and coconut tree digraphs under specific and arbitrary directional orientations.
Closed-Form Formulas for Fibonacci Numbers of Broom and Double Star Graphs Yudhi Yudhi; Eliana Neki; Fransiskus Fran; Raventino Raventino
ZERO: Jurnal Sains, Matematika dan Terapan Vol 9, No 3 (2025): Zero: Jurnal Sains Matematika dan Terapan
Publisher : UIN Sumatera Utara

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.30829/zero.v9i3.26839

Abstract

Let
Exploring the Metric Chromatic Number of Uniform, Centralized Uniform, and Cycle Uniform Theta Graphs Raventino Raventino; Fransiskus Fran
ZERO: Jurnal Sains, Matematika dan Terapan Vol 10, No 1 (2026): Zero: Jurnal Sains Matematika dan Terapan
Publisher : UIN Sumatera Utara

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.30829/zero.v10i1.27103

Abstract

Metric coloring allows adjacent vertices of a graph to share the same color provided that their associated distance vectors are distinct, leading to the concept of the metric chromatic number. This notion is closely related to problems of vertex distinguishability and resource allocation in network-like structures. In this paper, we present the first exact determination of the metric chromatic number for three families of theta type graphs: uniform theta graphs, centralized uniform theta graphs, and a newly introduced class called the cycle uniform theta graph, obtained by cyclically arranging uniform theta subgraphs. The proposed construction enables an investigation of how cyclic configurations influence metric coloring behavior. Using a constructive metric coloring approach, exact values of the metric chromatic number are obtained. It is shown that the uniform theta graph  and the centralized uniform theta graph  both satisfy  for all positive integers  and . For the cycle uniform theta graph , the metric chromatic number equals  when  and  have the same parity or when  is odd and  is even. In contrast,  when  is even and  is odd. This latter case arises because the longest path in the cyclic structure has odd length, forcing the graph to have chromatic number three. Since the graph is connected and its chromatic number is at most three, this structural constraint directly implies that three colors are also necessary for a valid metric coloring.
Exact Formulas for the Energy, Laplacian Energy, and Signless Laplacian Energy of Line Graphs of Lintang Graphs Fransiskus Fran; Meliana Pasaribu; Helmi Helmi; Raventino Raventino
ZERO: Jurnal Sains, Matematika dan Terapan Vol 10, No 2 (2026): Zero: Jurnal Sains Matematika dan Terapan
Publisher : UIN Sumatera Utara

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.30829/zero.v10i2.28777

Abstract

This study establishes exact formulas for the energy, Laplacian energy, and signless Laplacian energy of the lintang graph Ln and its line graph L(Ln). An algebraic spectral framework is employed to construct the associated matrices and determine their eigenvalues explicitly. In contrast to general results on line graph energies introduced by Ivan Gutman and subsequent studies, this work focuses on a specific two-hub graph structure for which explicit spectral formulas have not been previously reported. The results show that E(Ln) = 2 sqrt(2n), LE(L1) = QE(L1) = 10 sqrt(3), LE(Ln) = QE(Ln) = 4(n^2 - n + 2)/(n + 2) for n >= 2, and E(L(Ln)) = LE(L(Ln)) = QE(L(Ln)) = 4n - 4. These findings reveal a transition from sublinear to linear growth under the line graph transformation. The results contribute to spectral graph theory and are relevant to chemical graph theory and network analysis.
DIGRUNDY NUMBER OF DIRECTED STAR, BANANA TREE, FIREWORKS, AND COCONUT TREE GRAPHS Raventino Raventino; Fransiskus Fran
BAREKENG: Jurnal Ilmu Matematika dan Terapan Vol 20 No 4 (2026): BAREKENG: Journal of Mathematics and Its Application
Publisher : PATTIMURA UNIVERSITY

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.30598/barekengvol20iss4pp3213-3222

Abstract

A digrundy coloring of a graph is a vertex coloring in which every vertex assigned a higher color is adjacent to vertices assigned all smaller colors. The maximum number of colors that can be realized in such a coloring of an acyclic directed graph is called the digrundy number. This paper determines the digrundy numbers of several classes of acyclic directed graphs, namely directed star graphs, directed banana tree graphs, directed fireworks graphs, and directed coconut tree graphs. The analysis is based on structural properties of the graphs and combinatorial arguments derived from digrundy coloring constraints. The results show that the digrundy number of directed star graphs is under orientations where the central vertex satisfies and . For directed banana tree graphs with a specified orientation , the digrundy number is for and for . Under arbitrary orientations, directed fireworks graphs have digrundy number , while for directed coconut tree graphs , the digrundy number is bounded by These findings provide exact values of the digrundy number for the graph classes considered and highlight the role of structural constraints in governing digrundy coloring behavior.
On the Energy and Laplacian Energy of the Shadow,Thorn, and Middle Graphs of the Lintang Graph Fransiskus Fran; Meliana Pasaribu; Raventino Raventino; Tria Mutiara
(IJCSAM) International Journal of Computing Science and Applied Mathematics Vol. 12 No. 2 (2026)
Publisher : LPPM Institut Teknologi Sepuluh Nopember

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.12962/j24775401.ijcsam.v12i2.10744

Abstract

A graph with a transparent initial structure may exhibit markedly different spectral behavior after local or incidence-based transformations. This paper investigates the energy and Laplacian energy of the shadow, one-thorn, and middle graphs derived from the lintang graph Ln = K2 ∨Kn∼ = K2,n. The shadow graph is identified as D2(Ln)∼ = K4,2n, so its adjacency and Laplacian spectra are obtained from the standard spectra of complete bipartite graphs. The one-thorn graph T (Ln) is formed by attaching exactly one pendant vertex to each vertex of Ln, and its spectra are determined through quadratic transformations of the adjacency and Laplacian eigenvalues of the parent graph. The middle graph M (Ln) is analyzed using block matrices, the Schur complement, and rank-one determinant identities. The resulting energy formulas include E(D2(Ln)) = 4√2n and E(T (Ln)) = 2√2n + 4 + 2n, while exact expressions are also obtained for E(M (Ln)) and for the Laplacian energies of all three transformed graphs. In particular, LE(T (Ln)) = (4n/n+2) + 2√2(n−1) + √n2 + 4 + √n2 + 4n + 8. Moreover, D2(Ln) is integral precisely when n = 2k^2 for some positive integer k, whereas T (Ln) and M (Ln) are neither integral nor Laplacian integral for any n ≥ 2. These results show that neighborhood duplication, pendant extension, and incidence expansion produce distinct spectral and arithmetic effects, despite originating from the same complete bipartite base graph.
Packing Chromatic Number of Graph Families Constructed from the Fan Graph Raventino Raventino; Fransiskus Fran; Yudhi Yudhi
CAUCHY: Jurnal Matematika Murni dan Aplikasi Vol 11, No 2 (2026): CAUCHY: JURNAL MATEMATIKA MURNI DAN APLIKASI
Publisher : Mathematics Department, Maulana Malik Ibrahim State Islamic University of Malang

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.18860/cauchy.v11i2.44807

Abstract

Packing coloring is particularly sensitive to graph distances, since even a simple structural modification may change whether a color can be reused. This paper investigates the packing chromatic number of several constructions derived from the fan graph, namely the subdivision fan S(Fn), the umbrella graph Ur,s, two barbell-type fan constructions, and the generalized q-fan chain CFm(q) formed by sequentially joining the centers of q copies of Fm. The known packing chromatic number of the ordinary fan is used only as a reference result. By combining explicit packing colorings with lower bound arguments based on independence numbers, graph diameters, color class capacities, and cross-component distances, we prove that (S(Fn)) = 4 for n 2 and determine the exact packing chromatic numbers of the umbrella and both barbell-type constructions. For CFm(q), we establish general lower and upper bounds for all m, q 2 by showing that a color 2 can occur on at most q/( 1) vertices. Moreover, when m 2q 2, the bounds coincide and yield (CFm(q)) = q(m/2 + 2) t=1q1q/t. These results show that the packing chromatic behavior of fan-derived graphs is governed by the distance structure created by the underlying graph operation, particularly the extent to which low colors can be reused across different fan copies.
Rainbow Antimagic Coloring of Closed Sunflower Graphs and Their Multilayer Extensions Fransiskus Fran; Muammar Rifqi; Evi Noviani; Raventino Raventino
CAUCHY: Jurnal Matematika Murni dan Aplikasi Vol 11, No 2 (2026): CAUCHY: JURNAL MATEMATIKA MURNI DAN APLIKASI
Publisher : Mathematics Department, Maulana Malik Ibrahim State Islamic University of Malang

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.18860/cauchy.v11i2.46193

Abstract

The challenge in rainbow antimagic coloring is not to eliminate repeated edge weights, but to arrange them without obstructing rainbow paths. We address this problem for closed sunflower graphs and their multilayer extensions, with edge colors determined by sums of distinct vertex labels. Explicit bijective labelings yield upper bounds, while vertex degrees, distances, and regular subgraphs provide complementary lower bounds. The resulting estimates hold throughout the defining parameter ranges of both families, and the smallest closed sunflower graph has rainbow antimagic connection number seven. For cycle lengths of at least ten that are congruent to two modulo four, a separate construction reduces the gap between the bounds to two colors. In the multilayer family, an alternating labeling scheme gives an upper bound that grows linearly with the number of layers for each fixed cycle length and recovers the general closed sunflower construction in the two-layer case. All constructions include a complete count of distinct edge weights and a verification of rainbow connectivity. The general labelings admit rainbow spanning trees, whereas the improved construction uses explicit paths and fewer colors than a rainbow spanning tree would require. This distinction demonstrates how coordinated color reuse can strengthen constructive bounds beyond the spanning-tree approach.
Co-Authors Eliana Neki Evi Noviani Fransiskus Fran Fransiskus Fran Fransiskus Fran Fransiskus Fran Helmi Helmi Meliana Pasaribu Meliana Pasaribu Muammar Rifqi Nilamsari Kusumastuti Tria Mutiara Yeni Susanti Yudhi Yudhi Yudhi