Major General Hitchcock, Commissioner of Exchanges, is authorized and directed to offer Brigadier General Trimble, now a prisoner of war in Fort McHenry, in exchange for Major White, who is held as a prisoner at Richmond. He is also directed to send forward the offer of exchange by Henry M. Warfield, Esq. of Baltimore, under a flag of truce, and give him a pass to City Point.
Major General Hitchcock, Commissioner of Exchanges, is authorized and directed to offer Brigadier General Trumbull, now a prisoner of war in Fort McHenry, in exchange for Major White, who is held as a prisoner at Richmond.
It is also directed to send forward the offer of exchange by Henry M. Warfield, Esq. of Baltimore, under a flag of truce, and give him a pass to City Point.
Major General Hitchcock, Commissioner of Exchanges, is authorized and directed to offer Brigadier General T. R. M. Torrence, now a prisoner of war in Fort McHenry, in exchange for Major White, who is held as a prisoner at Richmond. He is also directed to send forward the offer of exchange by Henry M. Warfield, Esq., of Baltimore, under a flag of truce, and give him a pass to City Point.
Proof. Let $S$ be the generating set associated with $D$ as described in Proposition 2.5. By the circulant diagonalization theorem, the spectrum of $G_R(D) = \Gamma(R, S)$ is the multiset $\{\lambda_g\}_{g \in R}$ where
We remark that by Corollary 2.7, if $s \in R$ such that $Rs = I_i = Rx_i$ then $s$ has a unique representation of the form $s = ux_i$ where $u \in (R/Ann_R(x_i))^\times$ and $\hat{u}$ is a fixed lift of $u$ to $R^\times$. With this presentation, we can write
Here we recall that $\psi_i$ is the induced linear functional on $R/Ann_R(x_i)$. We conclude that $\lambda_g = \sum_{i=1}^{k} c(g, R/Ann_R(x_i))$. \hfill \square
The following corollary is simple yet important for our future work on perfect state transfers on gcd-graphs.
**Corollary 4.17.** Suppose that $g' = ug$ for some $u \in R^\times$. Then $\lambda_g = \lambda_{g'}$.
**Acknowledgements**
We thank the Department of Mathematics and Computer Science at Lake Forest College for their generous financial support through an Overleaf subscription. We also thank Ján Mináč for his constant encouragement and support.
**References**
1. Reza Akhtar, Megan Boggess, Tiffany Jackson-Henderson, Isidora Jiménez, Rachel Karpman, Amanda Kinzel, and Dan Pritikin, *On the unitary Cayley graph of a finite ring*, Electron. J. Combin. 16 (2009), no. 1, Research Paper 117, 13 pages.
2. Milan Bašić, Aleksandar Ilić, and Aleksandar Stamenković, *Maximal diameter of integral circulant graphs*, Information and Computation 301 (2024), 105208.
3. Maria Chudnovsky, Michal Cizek, Logan Crew, Ján Mináč, Tung T. Nguyen, Sophie Spirkl, and Nguyên Duy Tấn, *On prime Cayley graphs*, arXiv:2401.06062, to appear in Journal of Combinatorics (2024).
4. Thomas Honold, *Characterization of finite frobenius rings*, Archiv der Mathematik 76 (2001), no. 6, 406–415.
5. Irving Kaplansky, *Elementary divisors and modules*, Transactions of the American Mathematical Society 66 (1949), no. 2, 464–491.
6. Walter Klotz and Torsten Sander, *Some properties of unitary Cayley graphs*, The Electronic Journal of Combinatorics 14 (2007), no. 1, R45, 12 pages.
7. Erich Lamprecht, *Allgemeine theorie der Gaußschen Summen in endlichen kommutativen Ringen*, Mathematische Nachrichten 9 (1953), no. 3, 149–196.
8. Ján Mináč, Tung T Nguyen, and Nguyen Duy Tấn, *Isomorphic gcd-graphs over polynomial rings*, arXiv preprint arXiv:2411.01768 (2024).
9. ______, *On the gcd graphs over polynomial rings*, arXiv preprint arXiv:2409.01929 (2024).
Proof. Let $S$ be the generating set associated with $D$ as described in Proposition 2.5. By the circulant diagonalization theorem, the spectrum of $G_R(D) = \Gamma(R, S)$ is the multiset \{$\lambda_s$\}$_{s \in R}$ where
We remark that by Corollary 2.7, if $s \in R$ such that $Rs = I_i = Rx_i$ then $s$ has a unique representation of the form $s = u Rx_i$ where $u \in (R/\text{Ann}_R(x_i))^\times$ and $u \hat{i}$ is a fixed lift of $u$ to $R^\times$. With this presentation, we can write
Here we recall that $\psi_i$ is the induced linear functional on $R/\text{Ann}_R(x_i)$. We conclude that $\lambda_s = \sum_{i=1}^{k} c(g, R/\text{Ann}_R(x_i))$. \hfill \Box
The following corollary is simple yet important for our future work on perfect state transfers on gcd-graphs.
**Corollary 4.17.** Suppose that $g' = ug$ for some $u \in R^\times$. Then $\lambda_s = \lambda_{g'}$.
**Acknowledgements**
We thank the Department of Mathematics and Computer Science at Lake Forest College for their generous financial support through an Overleaf subscription. We also thank Ján Mináč for his constant encouragement and support.
**References**
1. Reza Akhtar, Megan Boggess, Tiffany Jackson-Henderson, Isidora Jiménez, Rachel Karpman, Amanda Kinzel, and Dan Pritikin, *On the unitary Cayley graph of a finite ring*, Electron. J. Combin. 16 (2009), no. 1, Research Paper 117, 13 pages.
2. Milan Bašić, Aleksandar Ilić, and Aleksandar Stamenković, *Maximal diameter of integral circulant graphs*, Information and Computation 301 (2024), 105208.
3. Maria Chudnovsky, Michal Cizek, Logan Crew, Ján Mináč, Tung T. Nguyen, Sophie Spirkl, and Nguyễn Duy Tần, *On prime Cayley graphs*, arXiv:2401.06062, to appear in Journal of Combinatorics (2024).
4. Thomas Honold, *Characterization of finite frobenius rings*, Archiv der Mathematik 76 (2001), no. 6, 406–415.
5. Irving Kaplansky, *Elementary divisors and modules*, Transactions of the American Mathematical Society 66 (1949), no. 2, 464–491.
6. Walter Klotz and Torsten Sander, *Some properties of unitary Cayley graphs*, The Electronic Journal of Combinatorics 14 (2007), no. 1, R45, 12 pages.
7. Erich Lamprecht, *Allgemeine theorie der Gaußischen Summen in endlichen kommutativen Ringen*, Mathematische Nachrichten 9 (1953), no. 3, 149–196.
8. Ján Mináč, Tung T. Nguyen, and Nguyen Duy Tần, *Isomorphic gcd-graphs over polynomial rings*, arXiv preprint arXiv:2101.01768 (2024).
9. ______, *On the gcd graphs over polynomial rings*, arXiv preprint arXiv:2409.01929 (2024).
Proof. Let $S$ be the generating set associated with $D$ as described in Proposition 2.5. By the circulant diagonalization theorem, the spectrum of $G_R(D) = \Gamma(R, S)$ is the multiset $\{\lambda_g\}_{g \in R}$ where
We remark that by Corollary 2.7, if $s \in R$ such that $Rs = I_i = Rx_i$ then $s$ has a unique representation of the form $s = ux_i$ where $u \in (R/\text{Ann}_R(x_i))^\times$ and $\hat{u}$ is a fixed lift of $u$ to $R^\times$. With this presentation, we can write
Here we recall that $\psi_{x_i}$ is the induced linear functional on $R/\text{Ann}_R(x_i)$. We conclude that $\lambda_g = \sum_{i=1}^{k} c(g, R/\text{Ann}_R(x_i)).$ $\square$
The following corollary is simple yet important for our future work on perfect state transfers on gcd-graphs.
Corollary 4.17. Suppose that $g^u = ug$ for some $u \in R^\times$. Then $\lambda_g = \lambda_{g^u}$.
Acknowledgements
We thank the Department of Mathematics and Computer Science at Lake Forest College for their generous financial support through an Overleaf subscription. We also thank Ján Mináč for his constant encouragement and support.
References
1. Reza Akhtar, Megan Boggess, Tiffany Jackson-Henderson, Isidora Jiménez, Rachel Karpman, Amanda Kinzel, and Dan Pritikin, On the unitary Cayley graph of a finite ring, Electron. J. Combin. 16 (2009), no. 1, Research Paper 117, 13 pages.
2. Milan Bašić, Aleksandar Ilić, and Aleksandar Stamenković, Maximal diameter of integral circulant graphs, Information and Computation 301 (2024), 105208.
3. Maria Chudnovsky, Michal Cizek, Logan Crew, Ján Mináč, Tung T. Nguyen, Sophie Spirkl, and Nguyễn Duy Tấn, On prime Cayley graphs, arXiv:2401.06062, to appear in Journal of Combinatorics (2024).
4. Thomas Honold, Characterization of finite frobenius rings, Archiv der Mathematik 76 (2001), no. 6, 406–415.
5. Irving Kaplansky, Elementary divisors and modules, Transactions of the American Mathematical Society 66 (1949), no. 2, 464–491.
6. Walter Klotz and Torsten Sander, Some properties of unitary Cayley graphs, The Electronic Journal of Combinatorics 14 (2007), no. 1, R45, 12 pages.
7. Erich Lamprecht, Allgemeine theorie der Gaußschen Summen in endlichen kommutativen Ringen, Mathematische Nachrichten 9 (1953), no. 3, 149–196.
8. Ján Mináč, Tung T Nguyen, and Nguyen Duy Tấn, Isomorphic gcd-graphs over polynomial rings, arXiv preprint arXiv:2411.01768 (2024).
9. ______, On the gcd graphs over polynomial rings, arXiv preprint arXiv:2409.01929 (2024).
Proof. Let $S$ be the generating set associated with $D$ as described in Proposition 2.5. By the circulant diagonalization theorem, the spectrum of $G_R(D) = \Gamma(R, S)$ is the multiset $\{\lambda_g\}_{g \in R}$ where
We remark that by Corollary 2.7, if $s \in R$ such that $Rs = I_i = Rx_i$ then $s$ has a unique representation of the form $s = ux_i$ where $u \in (R/\text{Ann}_R(x_i))^\times$ and $\hat{u}$ is a fixed lift of $u$ to $R^\times$. With this presentation, we can write
Here we recall that $\psi_x$ is the induced linear functional on $R/\text{Ann}_R(x_i)$. We conclude that $\lambda_g = \sum_{i=1}^{k} c(g, R/\text{Ann}_R(x_i))$. \qed
The following corollary is simple yet important for our future work on perfect state transfers on gcd-graphs.
**Corollary 4.17.** Suppose that $g' = ug$ for some $u \in R^\times$. Then $\lambda_g = \lambda_{g'}$.
**Acknowledgements**
We thank the Department of Mathematics and Computer Science at Lake Forest College for their generous financial support through an Overleaf subscription. We also thank Ján Mináč for his constant encouragement and support.
**References**
1. Reza Akhtar, Megan Boggess, Tiffany Jackson-Henderson, Isidora Jiménez, Rachel Karpman, Amanda Kinzel, and Dan Pritikin, *On the unitary Cayley graph of a finite ring*, Electron. J. Combin. **16** (2009), no. 1, Research Paper 117, 13 pages.
2. Milan Bašić, Aleksandar Ilić, and Aleksandar Stamenković, *Maximal diameter of integral circulant graphs*, Information and Computation **301** (2024), 105208.
3. Maria Chudnovsky, Michal Cizek, Logan Crew, Ján Mináč, Tung T. Nguyen, Sophie Spirkl, and Nguyễn Duy Tấn, *On prime Cayley graphs*, arXiv:2401.06062, to appear in Journal of Combinatorics (2024).
4. Thomas Honold, *Characterization of finite Frobenius rings*, Archiv der Mathematik **76** (2001), no. 6, 406–415.
5. Irving Kaplansky, *Elementary divisors and modules*, Transactions of the American Mathematical Society **66** (1949), no. 2, 464–491.
6. Walter Klotz and Torsten Sander, *Some properties of unitary Cayley graphs*, The Electronic Journal of Combinatorics **14** (2007), no. 1, R45, 12 pages.
7. Erich Lamprecht, *Allgemeine theorie der Gaußschen Summen in endlichen kommutativen Ringen*, Mathematische Nachrichten **9** (1953), no. 3, 149–196.
8. Ján Mináč, Tung T Nguyen, and Nguyen Duy Tấn, *Isomorphic gcd-graphs over polynomial rings*, arXiv preprint arXiv:2411.01768 (2024).
9. ______, *On the gcd graphs over polynomial rings*, arXiv preprint arXiv:2409.01929 (2024).
| Connects five fundamental constants (e, i, π, 1, 0), revealing the profound relationship between exponential functions and trigonometry. | Establishes that differentiation and integration are inverse operations. If F is an antiderivative of f, the definite integral equals F(b) - F(a). Revolutionized mathematical problem-solving. |
| In right triangles, the hypotenuse squared equals the sum of the squares of the other sides. Cornerstone of geometry with applications in navigation and architecture. | \(\nabla \cdot B = 0 \) |
| | Unified electricity and magnetism as manifestations of the same force. Describes electromagnetic field behavior, predicting waves traveling at light speed. Enabled technologies from radio to smartphones. |
A journey through the most elegant and influential formulas in mathematics
1. Euler’s Identity
\[ e^{i\pi} + 1 = 0 \]
Connects five fundamental constants (e, i, π, 1, 0), revealing the profound relationship between exponential functions and trigonometry.
2. Pythagorean Theorem
\[ a^2 + b^2 = c^2 \]
In right triangles, the hypotenuse squared equals the sum of the squares of the other sides. Cornerstone of geometry with applications in navigation and architecture.
3. The Fundamental Theorem of Calculus
\[\int_a^b f(x) \, dx = F(b) - F(a) \]
Establishes that differentiation and integration are inverse operations. If \( F \) is an antiderivative of \( f \), the definite integral equals \( F(b) - F(a) \). Revolutionized mathematical problem-solving.
Unified electricity and magnetism as manifestations of the same force. Describes electromagnetic field behavior, predicting waves traveling at light speed. Enabled technologies from radio to smartphones.
| Connects five fundamental constants (e, i, \(\pi \), 1, 0), revealing the profound relationship between exponential functions and trigonometry. | Establishes that differentiation and integration are inverse operations. If \( F \) is an antiderivative of \( f \), the definite integral equals \( F(b) - F(a) \). Revolutionized mathematical problem-solving. |
| In right triangles, the hypotenuse squared equals the sum of the squares of the other sides. Cornerstone of geometry with applications in navigation and architecture. | \[\nabla \cdot \mathbf{B} = 0 \] |
| Unified electricity and magnetism as manifestations of the same force. Describes electromagnetic field behavior, predicting waves traveling at light speed. Enabled technologies from radio to smartphones. |
| Connects five fundamental constants (e, i, \(\pi \), 1, 0), revealing the profound relationship between exponential functions and trigonometry. | Establishes that differentiation and integration are inverse operations. If \( F \) is an antiderivative of \( f \), the definite integral equals \( F(b) - F(a) \). Revolutionized mathematical problem-solving. |
| In right triangles, the hypotenuse squared equals the sum of the squares of the other sides. Cornerstone of geometry with applications in navigation and architecture. | \[\nabla \cdot \mathbf{B} = 0 \] |
| Unified electricity and magnetism as manifestations of the same force. Describes electromagnetic field behavior, predicting waves traveling at light speed. Enabled technologies from radio to smartphones. | |
| Connects five fundamental constants (e, i, π, 1, 0), revealing the profound relationship between exponential functions and trigonometry. | Establishes that differentiation and integration are inverse operations. If F is an antiderivative of f, the definite integral equals F(b) - F(a). Revolutionized mathematical problem-solving. |
| In right triangles, the hypotenuse squared equals the sum of the squares of the other sides. Cornerstone of geometry with applications in navigation and architecture. | $\nabla \cdot B = 0$ |
| | $\nabla \times E = -\frac{\partial B}{\partial t}$ |
| | Unified electricity and magnetism as manifestations of the same force. Describes electromagnetic field behavior, predicting waves traveling at light speed. Enabled technologies from radio to smartphones. |
| connects five fundamental constants (e, i, π, 1, 0), revealing the profound relationship between exponential functions and trigonometry. | In right triangles, the hypotenuse squared equals the sum of the squares of the other sides. Cornerstone of geometry with applications in navigation and architecture. |
| 3. The Fundamental Theorem of Calculus | 4. Maxwell's Equations |
| establishes that differentiation and integration are inverse operations. If \( F \) is an antiderivative of \( f \), the definite integral equals \( F(b) - F(a) \). Revolutionized mathematical problem-solving. | Unified electricity and magnetism as manifestations of the same force. Describes electromagnetic field behavior, predicting waves traveling at light speed. Enabled technologies from radio to smartphones. |
A journey through the most elegant and influential formulas in mathematics
1. Euler's Identity
\[ e^{i\pi} + 1 = 0 \]
Connects five fundamental constants (e, i, π, 1, 0), revealing the profound relationship between exponential functions and trigonometry.
2. Pythagorean Theorem
\[ a^2 + b^2 = c^2 \]
In right triangles, the hypotenuse squared equals the sum of the squares of the other sides. Cornerstone of geometry with applications in navigation and architecture.
3. The Fundamental Theorem of Calculus
\[\int_{a}^{b} f(x) \, dx = F(b) - F(a) \]
Establishes that differentiation and integration are inverse operations. If \( F \) is an antiderivative of \( f \), the definite integral equals \( F(b) - F(a) \). Revolutionized mathematical problem-solving.
Unified electricity and magnetism as manifestations of the same force. Describes electromagnetic field behavior, predicting waves traveling at light speed. Enabled technologies from radio to smartphones.
Connects five fundamental constants (e, i, π, 1, 0), revealing the profound relationship between exponential functions and trigonometry.
In right triangles, the hypotenuse squared equals the sum of the squares of the other sides. Cornerstone of geometry with applications in navigation and architecture.
| 3. The Fundamental Theorem of Calculus | 4. Maxwell's Equations |
Establishes that differentiation and integration are inverse operations. If F is an antiderivative of f, the definite integral equals F(b) - F(a). Revolutionized mathematical problem-solving.
Unified electricity and magnetism as manifestations of the same force. Describes electromagnetic field behavior, predicting waves traveling at light speed. Enabled technologies from radio to smartphones.
| Connects five fundamental constants (e, i, π, 1, 0), revealing the profound relationship between exponential functions and trigonometry. | In right triangles, the hypotenuse squared equals the sum of the squares of the other sides. Cornerstone of geometry with applications in navigation and architecture. |
| 3. The Fundamental Theorem of Calculus | 4. Maxwell's Equations |
| Establishes that differentiation and integration are inverse operations. If F is an antiderivative of f, the definite integral equals F(b) - F(a). Revolutionized mathematical problem-solving. | Unified electricity and magnetism as manifestations of the same force. Describes electromagnetic field behavior, predicting waves traveling at light speed. Enabled technologies from radio to smartphones. |
Connects five fundamental constants (e, i, π, 1, 0), revealing the profound relationship between exponential functions and trigonometry.
In right triangles, the hypotenuse squared equals the sum of the squares of the other sides. Cornerstone of geometry with applications in navigation and architecture.
| 3. The Fundamental Theorem of Calculus | 4. Maxwell's Equations |
Establishes that differentiation and integration are inverse operations. If F is an antiderivative of f, the definite integral equals F(b) - F(a). Revolutionized mathematical problem-solving.
Unified electricity and magnetism as manifestations of the same force. Describes electromagnetic field behavior, predicting waves traveling at light speed. Enabled technologies from radio to smartphones.
